Does the telecommunication “last mile” derive from the jail “last mile”?

The expression last mile appears to have been first used in a figurative sense in relation to capital punishment:

• the walk of a condemned person to the place of execution like the shaving of the head before the last mile —Saul Levitt (M-W)

It is not clear when the above usage started, but in more recent decades, apparently from the ‘70s, the expression started to be applied in completely different contexts such as technology and commerce:

Last mile:

(commerce) the last stage in a process, especially of a customer buying goods:

• One major problem of online shopping is "the last mile" - the delivery to the customer at their home.

(communicarions) the last stage in providing services such as cable television and broadband to homes and businesses:

• The firm offers an inexpensive and practical way of uncorking the bottleneck of the "last mile", bringing vast amounts of bandwidth to any business that wants it. (Cambridge Dictionary)

and, by extension, the expression has recently found new usages, as suggested by Wikipedia:

In recent years, usage of the term "last mile" has expanded outside the communications industries, to include other distribution networks....The term has also been used to describe education and training providers that more tightly link individuals with job opportunities.

According to Ngram the expression last mile is from the second half of the 18th century, but early usages appear to be mostly literal.

Questions:

• when did the usage of “last mile” in relation to capital punishment start? and did it originally refer to a real distance?

• is the later figurative usage (communications/commerce) related to the previous one or does it have a different origin?

As should be apparent from the common semantics of figurative (metaphorical) uses of the phrase 'last mile' — where the core meaning is "final period of greater difficulty, cost or suffering", which most definitely includes uses in commerce, telecommunications, transportation, and prison argot, among others — such uses are closely related.

In this sense ("final period of greater difficulty, cost or suffering"), the source of the metaphor seems to have been the tradition among religious pilgrims of walking the 'last mile' of their pilgrimage barefoot or otherwise in such a manner as to show their humility and devotion by increasing their suffering and hardship as they approached the end of their pilgrimage.

Although, considering the symbolic structure of a religious pilgrimage, the correspondence of the journey with a symbolic journey through life to death should be expected to have given rise to earlier figurative use, the earliest direct evidence of metaphorical use I could turn up on short notice was in John Donne's Holy Sonnets, sonnet VI:

This is my play's last scene, here Heavens appoint
My pilgrimage's last mile; and my race,
Idly yet quickly run, hath this last pace,
My span's last inch, my minute's latest point;
And gluttonous Death will instantly unjoint
My body and soul, and I shall sleep a space;
....

The composition date of the poem is not known. It is generally agreed, however, that the poem was composed in the first decade of the 1600s, and certainly it was composed before 1633, the date of its first publication.

While I cannot be precise about the date of the first metaphorical use of 'last mile' sponsored by the traditional practices of pilgrims making the last mile of their pilgrimage, it is probably no earlier than, and likely somewhat later than, the first attested use (possibly around 1225; possibly before 1200) of 'pilgrim' in this figurative sense from the OED:

Originally and chiefly in religious contexts: a person travelling through life, esp. one who undertakes a course of spiritual development leading towards heaven, a state of blessedness, etc.; a person who experiences life as a sojourn, exile, or period of estrangement from such a state.

To make all this exceedingly plain, the religious pilgrimage symbolically represents the journey through life to death, and so on to everlasting life; the last mile of the pilgrimage symbolically represents the spiritual and physical hardships attendant upon a nearer approach to death.

General metaphorical use of last mile in this sense was and continues to be commonplace. Specific applications of the sense can be found in prison argot, telecommunications, and transportation. It is no barrier to the relationship of the specific applications of the phrase in those areas with its source in the tradition of religious pilgrimage that contemporary users of the phrase are unaware of its distant origins.

Regarding the first of your two specific questions, I've found no definitive earlier use of the exact phrase 'last mile' in prison argot than the one mentioned by RaceYouAnytime, in the 1930 play by John Wexley titled The Last Mile. A similar phrase, the 'last milestone', does appear (perhaps coincidentally) in the 1920 biography of Charles Chapin, a newspaper editor convicted of his wife's murder and incarcerated in Sing Sing. About his incarceration he remarks

Looking backward now from the last milestone [in context, a direct reference to his incarceration in Sing Sing] of my life's journey, I am wondering what I might have achieved if I could have had the opportunities so many young men neglect.

Regarding the second of your two specific questions, that is,

is the later figurative usage (communications/commerce) related to the previous one or does it have a different origin?

see the full explication given in the first section of this answer.

I doubt the telecommunications usage of "last mile" has much to do with the figurative usage you cite.

The original telecommunications use of the "last mile" is very concrete: It's the pair of physical wires that run between the customer's house and the telephone branch office. It's not always exactly one mile long, of course, but in a typical urban setting a mile is a good ballpark indicator of the range of lengths we're talking about.

Here "last" has a primarily spatial meaning, rather than the temporal "last" in your prison sense. The last mile is the part of the communications chain that is closest to the actual customer, and therefore lies "last" when you view the world from the telco's head office.

I would date the rise of this expression not to the 1970s but specifically to 1982 when the Bell System was broken up by court order and the telephony market in North America suddenly had to support competition between different commercial carriers. Competitors to AT&T would be expected to install their own equipment in branch offices and lay their own cables to transmit calls between branch offices -- but for the last mile between the branch offices and every house it would be prohibitively expensive and disruptive if each new player on the market had to dig up every street everywhere to lay cables of their own to every house they wanted to have as customers. So making rules for how multiple competing carriers would share use of the same last-mile cables suddenly became important (and, in some cases, bitterly fought over).

A decade or two later those last miles became important for another reason -- namely with the spread of the Internet. Those miles-of-so of twisted copper wires were designed to carry a single voice conversation at a time, and it was and is a major engineering bottleneck to push a reasonable internet connection through them. Communications providers could upgrade their equipment at and between branch offices to newer technologies, such as optical fibers, but because each customer has his own last mile of cable it would be enormously costly to replace all of them with something better, and therefore most homes are still stuck with them.

This telecommunications usage has spawned its own figurative usages, where for example in Internet video distribution, the "last mile" often means the communication path between the end user and the distributor's server farm closest to that customer -- even though the server farm is probably not located in his local telephone exchange (though in some cases it may be).