At what point did "what a shame" come to mean "what a disappointment" or "what a missed opportunity"?
Problems with finding the requested point in time
Two things make answering this question rather difficult.
First, people have been using the expression "what a shame" for more than 400 years—often in such a way that the particular notion of shame embodied in the words is not immediately obvious. For example, Thomas Nashe, "The Foure Letters Confuted" (1592) contains this instance of the expression:
The parrat and the peacock have leisure to revive and repolish their expired workes. You speak like a friend : wele listen to you when you have repolished and expired your perfected degree. A Demy Doctor? what a shame it is!
Because this was written long before the "what a pity" sense of "what a shame" arose in English, we can confidently conclude that Nash intended the expression to convey the idea "what a source of shame it is." But if we were to read it in an uncertain context today, we might suppose that the sense of the phrase was along the lines of "what a regrettable [or unfortunate or disappointing] thing it is." In short, "what a shame" often functions as virtually a stand-alone expression whose precise meaning is difficult to infer, especially when read at a later date and without an intimate sense of what particular speakers habitually intended it to mean.
Second, most dictionaries do not to include dated definitions for the phrase "what a shame," which means that to estimate the origin date of "what a shame" in the sense of "what a pity" or "how disappointing," we must look instead for the first appearance of a dictionary definition of shame in the relatively mild sense of "disappointment or cause for regret." Of course, if the dictionary in question includes this definition in the first place because use of "what a shame" in the sense of "how unfortunate" is already prevalent, we will surely underestimate the age of such usage.
With those difficulties in mind, I propose to walk through various editions of Merriam-Webster's Collegiate Dictionary in search of the earliest entry for shame in the sense of "regrettable." This dictionary series is not especially brilliant, but it has the advantage of having come out in successive new editions in 1898, 1910, 1916, 1931, 1936, 1949, 1963, 1973, 1983, 1993, and 2003, creating a series of fairly narrow windows to consult. And since I have all eleven editions, I can identify to the decade when Merriam-Webster considered the usage sufficiently established to merit inclusion in the Collegiate Dictionary.
How different Collegiate Dictionaries treated 'shame' as a noun
Let's take a look at how MW's treatment of shame as a noun changed over the 105 years in question.
From the First Collegiate (1898):
Shame n. 1. A painful sensation excited by consciousness of guilt or improper action. 2. Reproach incurred or suffered; dishonor; ignominy. 3. Cause or reason of shame. 4. The parts required by modesty to be covered.
Second Collegiate (1910): no change.
From the Third Collegiate (1916):
shame n. 1. Painful feeling or emotion excited by consciousness of guilt or impropriety; also, susceptibility to such feeling. 2. Reproach incurred; dishonor; ignominy; contempt. 3. Cause of shame; a disgrace.
Fourth Collegiate (1931): no change.
From the Fifth Collegiate (1936):
shame n. 1. a Painful emotion excited by a consciousness of guilt, shortcoming, or impropriety. b Susceptibility to such feeling or emotion. 2. Disgrace; dishonor. 3. That which brings discredit or reproach; as, it's a shame.
Sixth Collegiate (1949): no change.
From the Seventh Collegiate (1963):
shame n. 1 a : a painful emotion caused by a consciousness of guilt, shortcoming, or impropriety b : the susceptibility to such emotion 2 : DISHONOR, DISGRACE 3 a : something that brings strong regret, censure, or reproach b : a cause of feeling shame
Eighth Collegiate (1973): no change, except to definition 2:
2 : a condition of humiliating disgrace or disrepute : IGNOMINY
Ninth Collegiate (1983): no change.
Tenth Collegiate (1993): no change, except to definition 3a:
3 a : something that brings censure or reproach; also : something to be regretted : PITY {it's a shame you'll miss the party}
Eleventh Collegiate (2003): no change, except the addition of examples of senses 1b and 2, and a minor alteration of the example of sense 3a (to "it's a shame you can't go").
The emergence of a meaning of shame along the lines of "cause of disappointment or regret" seems to have occurred in stages, at least by Merriam-Webster's telling. The first hint that it was emerging appears in the 1963 Seventh Collegiate's reference to "something that brings strong regret, censure, or reproach." But the major breakthrough came with the 1993 Tenth Collegiate's simultaneous weakening of "strong regret, censure, or reproach" to "simple censure or reproach" and it's introduction of "something to be regretted : PITY." There, in full bloom, is the shame of "what a shame" in its modern sense.
Historical Examples
It seems highly likely, however, that people were saying "what a shame" in the sense of "what a pity" long before 1993. Consider this bit of dialogue from Robert Kissell, "The Big War," in Cincinnati Magazine (October 1972):
"It's a shame, you know," Eric Russell said.
"What's a shame?"
"That you can't hear about war from the people who got killed."
"What a positively inane remark!" Stowe replied.
"Harrelson lost a leg," Ashton said.
"He fell out of a jeep in the Aleutians. Dead drunk," Russell said.
"Which was still closer than you got to anything," Stowe answered.
From the context of these remarks, it appears that Russell doesn't mean "It's a shame" in the sense of "It's a cause of dishonor or disgrace or reproach or discredit," but rather in the sense of "It's a source of regret" or "It's a pity."
But much the same can be said of this instance from "Has a Kangaroo," in the "Young Folks' Letters" section of the Longreach [Queensland] Leader (October 12, 1940):
You seem to have quite a few pets, "Rainbow." I am pleased that you like the singing so much, especially our own girls. What a shame that your brother's dog died. I bet he misses it. I hope your father has recovered by now. That seems a good way to catch fish.
The shame here, too, is "pity" or "cause for regret" rather than "ignominy."
This instance from "London Literary Notes," in the New York Tribune (February 18, 1920) is not far off the modern mark, either:
John Hastings Turner told me his book "Simple Souls," was based on his play of the same name. I asked why his play hadn't been done. He said they couldn't get a theater for it. I said what a shame that a play like "Simple Souls" should have to wait when so many rotten revues were filling the theaters. Turner laughed at this, and I discovered why later on; he's the sole author, part author, dialogue writer or filler-in of half the revues in London, and has made a pot of money at it.
You can make a case that the author of this last instance meant "what a shame" in the sense of "how shameful"—but it is easy to see how such expressions might grade into the sense of "what a pity" over time, given that "how shameful" is rather an exaggerated response to someone's not being able to get his play put on because there were no open theaters.
And this instance, from "The End of the Season," in the [Estes Park, Colorado] Trail Talk (September 3, 1920) can only reasonably be interpreted as meaning "what a pity":
What a shame that the tourists cannot stay to enjoy the month of September, one of the most glorious of the year, in the Rocky Mountain wonderland.
Jumping back another 20 years, we find examples like this one, from "In the Domain of Music" (an interview with 87-year-old Giuseppe Verdi) in the St. Paul [Minnesota] Globe (November 4, 1900):
"And Wagner?"
"Wagner, too. Wagner first for the operatic stage. Music would have been very different if he had not been there. But to me, personally, Brahms as a pure musician was dearer. He was deep and yet comprehensible. What a shame that he did not write an opera."
Clearly Verdi considers Brahms's failure to compose an opera a source of regret, not a stain on Brahms's memory.
Conclusions
Merriam-Webster will tell you that shame in the sense of "something that brings strong regret" dates to no later than 1963 and that shame in the sense of "something to be regretted : PITY" dates to no later than 1993. But these dates represent the points at which Merriam-Webster decided no longer to ignore the meanings that it acknowledged in editions of the Collegiate Dictionary published in those years; the meanings themselves go back decades earlier, as the examples noted above indicate.
It thus appears that use of "what a shame" to mean "how regrettable" or "what a pity" goes back at least to the dawn of the twentieth century, although dictionaries may not have ratified the usage until much later.