Why is a cold shower called a “Scottish Shower”?

Solution 1:

Note: This answer presents research relevant to the poster's question "When and where did the expression originate?" It does not address the questions "Why is a cold shower called a 'Scottish Shower'?" and "Has it really something to do with a traditional Scottish custom or is it just a literary invention?"—for the simple reason that I couldn't find any reliable information on those issues. For compelling evidence on those points, I commend JEL's excellent answer to your attention.

Evidently, alternating hot and cold streams of water during a shower or spa treatment has been known for at least 140 years as a "Scotch [or Scottish] shower bath." From "Dubois on Treatment of Neuralgia by Hydrotherapy and Electricity Combined," first published in the London Medical Record (April 15, 1879):

In sciatica the following treatment has proved most successful in a case where the patient had been suffering for two years, without being able to obtain any relief. In the morning the hot-air bath was given, and followed on alternate days by a cold shower bath, a Scotch shower bath being given on the other days. At night the continuous current was applied, applying the positive pole to the lumbar region, and the negative, first to the nates, then to the popliteal region.

From Jean Dardel, "The Thermal Treatment of Aix-les-Bains," in International Clinics: A Quarterly of Clinical Lectures (1907):

The massage being terminated, the patient places himself in a corner of the cabin, and there receives the douche in full, or in a shower, or in a sprinkle, which is the most important of all. The last part is extremely important, and the doctors know how to obtain different results, from the use of a broken jet or from a full jet, the jet falling on the patient like a spout, throwing out in full force the water upon him, or coming gently upon him like a shower of rain, either in cold or in warm showers or in Scotch shower-baths or douches.

The earliest reference to a "Scotch shower bath" that I could find in a search of the British Newspaper Archive appears as a passing reference in the [London] Morning Post (September 6, 1861). The OCR rendition of this instance is very poor (and I don't have a BNA subscription that would permit me to check the actual photocopy of the article), but it probably reads approximately as follows:

At the casino we find a gymnasium directed by a clever professor: hot sea-water baths, and fresh-water baths, with ordinary or Scotch shower baths: a breathing-hall for the sea pulverised water, a good treatment for laryngial sickness; a pistol aiming and billiard hall, with all sorts of games : reading-rooms, where all newspapers are to be found ; drawing-rooms for the pianoforte, with a distinguished professor specially engaged for the establishment; ...

The exact meaning of "Scotch shower bath" may have changed through the years, but this description, from Robert Maigne & ‎Walter L. Nieves, Diagnosis and Treatment of Common Pain of Vertebral Origin: A Manual Medicine Approach (1996) [combined snippets] suggests that it is still in therapeutic use:

The "Scotch" shower is good in sciatica, and it is performed as follows: full jet at 39°C for 6–8 minutes, followed by a cold shower at 1.1°C in broken jet for 30 seconds. The feet are always kept warm.

An instance from 1938 indicates that "Scotch shower-bath" was understood in similar terms at that date. From an unidentified article in The Annals of Otology, Rhinology & Laryngology (1938) [combined snippets]:

Moreover, some of these patients cannot endure a cold shower-bath, and may suffer collapse. However, this defective vasomotor function may, in a certain degree, be improved by cautious administration of alternate, warm and cold shower-baths (Scotch shower-bath), or by the patient's standing with his feet in warm water. Besides, it is one of the most important rules that a Ménière patient must never feel cold, whether during treatment or otherwise.

Victor Cherbuliez, Meta Holdenis: A Novel (1887) briefly describes a Scotch shower-bath from the patient's perspective:

Did you ever take a Scotch shower-bath, madame? Do you know how the unfortunate bather, who has just been plunged into hot water, feels, as the first ice-cold drops of the shower-bath above run down his shoulders?

The term appears figuratively in a press release from the Soviet Embassy in London on August 10, 1943, reprinted in Soviet War News (1943):

“The battle of Orel has no parallel," Radio Breslau on August 6. "There was no battle whatever at Orel," quoth Berlin radio on the same day. "Orel is the greatest defeat the Russionas have suffered" (Radio Berlin, August 6). "We do not intend to try to sweeten the bitterness of the situation" (Radio Donau, August 6).The Fritzes are welcome to make head or tail of it if they can. German comment on Orel is like a Scotch shower, scalding hot and ice-cold alternately. They kept mum about Byelgorod.

Evidently, figurative use of "Scotch shower" to indicate "hot and cold" political treatment has made the expression idiomatic in Italy (as doccia scozzese) and France (("le regime de la douche ecossaise"). The instance from France, reported in Time magazine (1949) is interesting because it views the treatment as being prompted by a shortage of hot water, rather than as being a designed therapeutic process:

Lange's Norway, they said, had followed just that procedure. And when a little country like Norway takes its stand clearly, they added, it should certainly not get from the U.S. what a Quai d'Orsay official colloquially called the Scotch shower treatment ("le regime de la douche ecossaise"—intermittent hot and cold water, to save fuel.

An article in The Critic (1972) [combined snippets] reports that Mussolini was a master of the figurative "Scotch shower":

Benito Mussolini, fundamentally a believer in force, combined the stick with the carrot of conciliatory gestures, charm and flattery, or as they say in Italian, "the Scotch shower," alternating applications of hot and cold. He had an excellent sense of timing and the gift of quickly summing up a political situation.

On a more literal—and far less relevant—note, Punch (November 1851) offers this joke (presumably) about the mistiness of Scottish showers:

A Conundrum Made by a Little Boy Only Seven Years Old.—Why is an umbrella like a Scottish shower? Because the moment it rains it's missed.

Conclusions

The historical line form James Bond's hot-and-cold shower extends, with a fair degree of continuity, back to at least 1861, although the phrase "Scottish [or more often, Scotch] shower[-bath]" is especially common in continental European accounts (from 1879 and later) of hydrotherapies involving exposure to spouts or jets of alternately hot and cold water. Accounts separated by more than 115 years endorse "Scotch showers" as treatment for sciatica.

As a figurative expression, "Scotch shower" seems to have gained some currency (in translation) in France, Italy, and the Soviet Union, where it referred (and may still refer) to behavior or policy that in U.S. English might more likely be referred to instead as "running hot and cold." There is even a word in Esperanto for "a shower that alternately runs cold and scalds": "Škoda duso—a 'Scottish shower'," according to a 2010 article in the Michigan Quarterly Review [the relevant snippet does not appear in the snippet window].

Solution 2:

Uses of the phrases 'Scotch douche', 'Scotch shower', and 'Scottish shower' in English, which are in evidence in print as early as 1839, may derive from earlier uses of the equivalent French phrase, douche écossaise (literally, "Scottish shower"), in medical-therapeutic contexts.

The French phrase, which at first refers to an alternately hot and cold shower, shows up as early as 1825, in Revue encyclopédique, ou Analyse raisonnée:

C'est dans cette salle que M. Despine, habile médecin, a établi la douche écossaise, dont il a obtenu de si heureux résultats dans le traitement des affections nerveuses.

[It is in this room that M. Despine, a skilful physician, has established the Scottish shower, from which he has obtained such happy results in the treatment of nervous affections.
— Translation from Google Translate.]

The French douche écossaise in later use has, in addition, a more figurative sense of a "series of ups and downs; roller-coaster ride" (translation of douche écossaise from Collins French to English dictionary). This later French sense is remarked by Harold Ettlinger in The axis on the air, 1943 (emphasis mine):

At that time too we saw the strategy of confusion worked so effectively on the French. We saw them subjected to what they called "a Scottish shower," alternately hot and cold, until people did not know what to think or believe, and until the majority reaction after Munich was a kind of benumbed relief.

The English phrase 'Scotch douche' appears in print as early as 1839 in connection with mineral water treatment for facial neuralgia (tic doloreux), in The Medical Examiner, Philadelphia, June 29 (emphasis mine):

Same waters taken internally, in baths, and Scotch douches upon the head....

"Same waters" refers to the waters "of Ussat, of St. Sauveur, of Salut at Bagneres de Bigorre, of Neris, Bains, &c.," all French hydrotherapy sites.

'Scotch douche' appears again in a descriptive paragraph in the July, 1847 issue of The American Journal of the Medical Sciences:

The shower bath is familiar to every one; but there is one form, often called the "Scotch douche," which is not much known. In this the shower is alternately cold and warm, causing rapid changes in the cutaneous circulation without much shock; it is used when reaction is feeble, as a preparative for the cold douche.

The first evidence of the exact phrase 'Scotch shower', in the sense of "alternately hot and cold shower", that I could find in English was in a series of identical advertisements for the "Sea Baths at Fécamp". Fécamp is in France, across the English Channel from Brighton, and so convenient to London. The ads appeared weekly from 10 May through 28 June 1861, then again 09 September through 26 September 1861, in The Morning Post, London, England (paywalled, 10 May example linked):

At the casino we find a gymnasium directed by a clever professor: hot sea-water baths, and fresh-water baths, with ordinary or Scotch shower baths....

A nearly identical version of this ad appeared in the 02 July 1861 edition of The Times (London; paywalled).

The first evidence I found in print of the exact phrase 'Scottish shower' (in the sense of "alternately hot and cold shower") was the 1943 use by Ettlinger, cited above with reference to figurative use of the French phrase.

English lexical translations of the French phrase show up in print at least as early as 1868, in the Medical lexicon...a French as well as English Medical Lexicon by Robley Dunglison, with particular reference to use of the therapeutic technique of alternating hot and cold showers:

DOUCHE ÉCOSSAISE, Douche, transition — d. Scotch, Douche, transition.
DOUCHE, TRANSITION, Scotch douche, (F.) Douche Écossaise. A douche, which consists in the successive use of hot and cold water.

Dunglison, a US physician, intends a specialized sense of the French douche (op. cit.):

This term is applied to a column of fluid, of a determinate nature and temperature, let fall upon the body. ... They may be cold or warm, according to circumstances. ... The Douche communicates a considerable and peculiar shock to the nervous system; and is one of the most successful means for taming the furious maniac.

Although the source of the French phrase douche écossaise, from which the English phrase appears as a matter of evident history to derive, must be a matter of speculation for me, a likely origin is suggested by prominent earlier (pre-1825) medical literature relating to the therapeutic use of cold water.

In 1805, the Scottish physician James Currie published an extensive review on the therapeutic use of cold and warm water in medicine. The work was titled Medical Reports, on the Effects of Water, Cold and Warm.

In Chapter 1 of his seminal work on the subject, Currie quotes a medical report by Dr. William Wright ("president of the College of Physicians, Edinburgh", op. cit., p. 1), another Scottish physician, on the treatment of fever with cold water ablutions (op. cit. pp. 3-4):

When upon deck, my pains were greatly mitigated, and the colder the air the better....
...three buckets full of salt water were then thrown at once upon me; the shock was great, but I felt immediate relief...the same febrile symptoms threatened a return, and I had again recourse to the same method as before, with the same good effect....
Sept. 10th...used the cold bath twice.
Sept. 11th...to prevent a relapse, I used the cold bath twice.

Wright goes on to describe the successful treatment of another patient with the same method.

Currie concludes the chapter thus (op. cit., p. 5):

  To this interesting narrative, Dr. Wright adds some general observations on the traces that are to be found of the use of cold water internally and externally in fevers, in several works ancient and modern....
  Having before experienced that Dr. Wright was a safe guide, I...determined on following his practice...I was further encouraged, by learning, that my respectable colleague, Dr. Brandreth, had employed cold water externally in some recent cases of fever with happy effects.

It is probable that Currie's influential work, itself based on the experiences and innovative practices of what might be called the Scottish medical contingent, prompted the French adoption of douche écossaise as the medical term of art for alternately warm and cold showers.