According to Don Wilmeth, The Language of American Popular Entertainment: A Glossary of Argot, Slang, and Terminology (1981), the general term for a solo comedic performance in vaudeville was "talking single":
Talking single: A one-person vaudeville act using for material stories, gags, and the like. A kind of early stand-up comic.
In instances where the comic uses an actual telephone in the act, the telephone is a prop, short for property and the performance might be called a "prop gag." Here is Wilmeth's entry for that term:
Prop gag: A comic routine depending on a property [where property has the meaning "stage accessory or object handled by a performer on-stage in vaudeville, scenery excepted, as in legitimate theatre"].
Of course, many standup comics do their routines without a prop, positioning their hand as if it were holding a real phone.
The most common term I've found for an act consisting of a solo performer carrying on an imaginary telephone conversation is telephone monologue. This term, which applies to both serious and comic performances, is at least 110 years old. From "Methodist Bazaar," in the Jackson [Missouri] Herald (December 26, 1907):
In the first night entertainment Col. T. W. Birmingham proved to the audience that he was at home on the stage and he rendered the Long Distance Telephone Monologue to perfection.
From "Improvement Clubs German Entertainment a Grand Success," in the [San Rafael, California] Marin County Tocsin (June 22, 1912):
Sam Barclay kept the audience in continual laughter by his telephone monologue, and the same was received with such generous applause that he responded with a few illustrations of the sayings of noted San Francisco characters.
And from "The English Pierrots," in the [Perth, West Australia] Daily News (June 9, 1916):
Mr. Charles Lawrence was heard in some of his delightful "Pianologues," one of his most entertaining numbers being his well-known telephone monologue, "Wrong Number!"
Two standup comics of the 1950s and later are famous for performing this type of act: Shelly Berman and Bob Newhart. In his memoir, I Shouldn't Even Be Doing This!: And Other Things That Strike Me as Funny (2006), Newhart refers to the shtick as "a telephone routine" or "my one-sided telephone conversation." Elsewhere in the book, he refers to the telephone as "my partner."
Sarah Linn, "Calling Bob Newhart: The comedy legend talks about sitcoms, censorship and sex," in the [San Luis Obispo, California] Tribune (January 17, 2017) also uses the term "one-sided telephone conversations," so that may be an alternative category name to "telephone monologue." Another possibility is "imaginary phone act."
As noted in the Wikipedia article on Shelly Berman (cited above), George Jessel incorporated a one-sided telephone conversation in his stage act may have been in the 1920s and 1930s. Mark Rothman of markrothmansblog has an interesting two-part discussion of the history of "imaginary phone" monologue sketches and the personalities of the comics who performed them, titled "The Telephone Hour" and "The Telephone Hour 2". Rothman credits Jessel (starting in the 1920s), Arlene Harris (in the 1930s and 1940s), and Betty Walker (in the 1950s) as performing similar acts. But as the newspaper examples noted earlier in this answer make clear, news accounts of telephone monologues go back to 1907 at least.