To answer the when part of your question, it looks like the nickname took hold in the early 1960s. The earliest use of it in print I could find is from a 1962 television trade publication:
For such of the faithful who do care, Mike Dann, CBS-TV's vice president of network programs, has some happy thoughts. 'I think the boys are about to have their turn on the tube,' Dann cheerfully predicts.
Television magazine, Volume 19, Issue 3
It appears the term was used widely by advertisements for televisions promoting the latest technology behind their "color picture tubes." The term's popularity increased greatly in the 70s, peaked in the 80s, and has been in decline since then. (Unless, of course, you count the use of YouTube.)
In response to Jacinto's comment, I searched again and found a reference to a man being a wallflower, in which it was not so gender-specific, although in the context it is already known that they are talking about a man. In 1913, Corra Harris wrote In Search of a Husband. On pages 47-48 I found the following passage of conversation between two women:
"Still it was queer, not to introduce him," I insisted
"Not queer, merely impudent. But he had the best of it. The difference between being a wallflower and an indifferent celebrity upon such as occasion is accomplished by turning your back to the company. Wallflowers sit or stand face forward, confessing defeat. He turned his back on us and put us out of countenance."
I've located a couple of other early 20th century references to male wallflowers, which are much more gender-specific.
From 1910, in At the home plate, by Albertus T. Dudley, page 123:
"I wonder what kind of a kid Crusty was when he was at school!" he thought, as he staggered back to his seat. "I'll bet he was a wallflower. It's queer that he should have a brother who can play football."
From March 1918, in The Recruit. A pictorial naval magazine. v.4 no.3 at the end of the first paragraph of page 24, from a short story called "The Wall Flower":
He was a wallflower
When there was a happy crowd grouped enthusiastically about the piano in one of the girls' homes, singing, laughing, and at ease, where was he? Sticking out as solemn and aloof as a sore thumb, that's where.
Further back in 1884/5, in The Freemason's repository. v.14 (following a list of questions regarding active participation in the Freemasonry), on page 133:
Is it not a lamentable fact that in the great majority of instances these questions would have to be answered in the negative? And now, by virtue of his office, he is entitles to a seat in the Grand Lodge. Is it any wonder he is a wall-flower there. Is it strange that the business of the Craft falls into the hands of a few men, who at the sessions of the Grand Bodies are overwhelmed with work?
Best Answer
As a noun meaning "native or citizen of the island of Lesbos," Lesbian has been used in English as a noun since the earliest translations of Herodotus and Thucydides. As a result, the notion of using lesbian as a noun when the later meaning of "a woman who is a homosexual" (Merriam-Webster's current definition) arose, around 1890, would not have struck many hearers as being a startling innovation. Interestingly, MW reports that both the noun lesbian and the noun sapphism (meaning "lesbianism") around 1890.
Though Sappho was well known to have been a native of Lesbos, one eighteenth-century makes no reference to her in expressing an unflattering opinion as to the licentiousness of the island's population (most particularly the men). From An Universal History, from the Earliest Account of Time to the Present, volume 3 (1738):
I haven't found a picture of Goltzius's medal online; but another author observes that the medals of Lesbos were notably oriented toward sexuality. From Richard Knight, The Worship of Priapus and Its Connection with the Mystic Theology of the Ancients (1865):
William Guthrie, A New Geographical, Historical, and Commercial Grammar (1815) recounts Lesbos's past and continuing claims to fame:
And Julius Rosenbaum, The Plague of Lust: Being a History of Venereal Disease in Classical Antiquity (1901) reports that "irrumation" (oral sex performed on a man) was associated in ancient Greece with the island of Lesbos:
The connection between Sappho and the modern sense of lesbianism appears not to have been universally recognized in the nineteenth century. From Agnes Smith, Olympus and Its Inhabitants (1851):
On the other hand, William Mure, A Critical History of the Language and Literature of Antient Greece, volume 3 (1854) alludes to the sexual attraction of women to other women in connection with "the Lesbian vice":
Mure's discussion in an appendix dedicated to a discussion of "the Lesbian vice" is especially interesting because in it he makes clear that his focus there is not on what Rosenbaum in 1901 calls "the Lesbian mode" of sexual interaction between men and women.
The earliest unmistakable instance of "a lesbian" in the modern sense of the term that a Google Books search returns is from Edgar Saltus, The Imperial Orgy: An Account of the Tsars from the First to the Last (1920):
This is, of course, much later than the earliest citation in Merriam-Webster, and is undoubtedly a sign of the limitations of the Google Books database or search engine or both. The earliest Google Books match for lesbians is from Jacobus X, Crossways of Sex: A Study in Eroto-pathology, volume 2 (1904), which devotes a lengthy chapter to "Lesbians, Tribades, Fricarelles, and Saphists."