The phrase not Pygmalion likely is a euphemistic, jocular variant of not bloody likely, itself emphatic for not likely.

The phrase in use—by Kerry Myers, Letters Editor, in The Sydney Morning Herald (Sydney, New South Wales, Australia) of 28^th June 1999:

Local bands out of step? Not Pygmalion likely, according to many, many brassed-off readers who wrote last week, outraged over the decision by SOCOG¹’s Rich Birch to import American and Japanese bands to perform at the opening ceremony of the Olympic Games.

¹ SOCOG: Sydney Organising Committee for the Games of the XXVII Olympiad, i.e., for the 2000 Summer Olympics

The phrase not Pygmalion likely originated in the sensation caused in 1914 by the use of the expletive bloody (which was taboo at that time) in not bloody likely by the character of Eliza Doolittle in Pygmalion, by the Irish playwright, critic, polemicist and political activist George Bernard Shaw (1856-1950).

Pygmalion was produced in London at His Majesty’s Theatre, opening on 11^th April 1914 and running for 118 performances. The English actress Mrs Patrick Campbell (Beatrice Stella Tanner – 1865-1940) interpreted the role of Eliza Doolittle; the English actor and theatre manager Herbert Beerbohm Tree (1852-1917) played Henry Higgins.

This is what G. S. Street, of the Lord Chamberlain’s Office, which was responsible for theatre censorship, wrote on 23^rd February 1914 about Pygmalion and Eliza’s use of the word bloody:

The Pygmalion of the Play is one Higgins, a professor of Phonetics who boasts that in a short time he can turn anyone into anything so far as speech goes. He wagers with his friend, a Colonel, that he can make Liza, a flower girl, pass for a duchess at an ambassador’s garden party. The first three acts are taken up with preliminaries and education. An amusing character, the girl’s father, boasts himself one of the “undeserving poor” and preaches a diverting morality. In the fourth act, Liza is transformed into a lady as [?] speech and manners and has won Higgins his triumph. But he and the Colonel have reckoned without her feelings. They have all lived together– all this is quite innocent, of course– and she is outraged by their discussing the success of the experiment as though she was an automoton [sic] and not a human being. She runs away and in the fifth act they find they cannot do without her: in the end she consents to come back and it is clear will be treated humanly as a daughter or sister. She, of course, is Galatea².
The Play is entirely without offence, except perhaps to the opinions of old-fashioned people who must be accustomed to having their opinions offended in modern dialogue. I notice, however, one detail. On Page 46, the word “bloody” slips out of the as yet only partially educated Liza and on the next page a silly young woman uses it under the impression that it is part of the new “small talk”. The word is not used in anger, of course, and the incident is merely funny. I think it would be a mistake to be particular about it, but since the word has been forbidden in other plays– in a different sort of connection, however– I mention it.
Recommended for license.

² In Greek Mythology, Galatea was the name given to the statue fashioned by Pygmalion and brought to life.

This is the passage of Pygmalion in which Eliza utters not bloody likely—from Pygmalion: a Play in Five Acts: by a Fellow of the Royal Society of Literature. Rough Proof—Unpublished (London: Constable and Company Ltd., 1913):

Higgins [looking at his watch] Ahem!
Liza [looking round at him; taking the hint; and rising] Well: I must go. [The men rise, except Higgins. Freddy goes to the door]. So pleased to have met you. Good-bye. [She shakes hands with Mrs Higgins].
Mrs Higgins. Good-bye.
Liza. Good-bye, Colonel Pickering.
Pickering. Good-bye, Miss Doolittle. [They shake hands].
Liza [nodding to the others] Good-bye, all.
Freddy [opening the door for her] Are you walking across the Park, Miss Doolittle? If so—
Liza. Walk! Not bloody likely! [Freddy reels: Higgins collapses on the divan]. I am going in a taxi. [She goes out].

On 19^th April 1914, the St. Louis Post-Dispatch (St. Louis, Missouri, USA) published these interesting explanations:

London, April 18.—[…]
[…]
Mrs. Campbell is playing the leading role in Bernard Shaw’s new piece, “Pygmalion,” which has aroused the indignation of the ultra-respectable critics because of the introduction in Mrs. Campbell’s speeches of swear words used only by the lowest classes of society.
The criticisms of the use of the adjective “bloody” in “Pygmalion” has provoked from Shaw a loquacious reply in the Daily News. He points out that the word as used by Eliza Doolittle, the slum-bred flower girl, is in common use as an expletive among four-fifths of the English nation, adding:
“I was not the first to use this word on the stage, but I was the first to see that Mrs. Patrick Campbell, in the part of Eliza, could bring down the house with it quite inoffensively by the very simple artistic stroke of using it naturally and sincerely.”
Shaw argues that by making one of the other characters—“a would-be smart lady”—repeat the word he has provided a strong antidote to Mrs. Campbell’s utterance, thus preventing “all smart London ‘bloodying’ all through the season.”
As a result of the newspaper publicity given to the play, London’s latest catch phrase is “Not Pygmalion likely.” A Cockney coster would say: “Not bloody likely.”

The earliest actual use of not Pygmalion likely that I have found is from The Sporting Times (London, England) of 30^th May 1914:

Poor old Masterman! The birds in the air have nests, the beasts in the field their lairs—and this “pure and beautiful soul” cannot find so much as a blessed seat for himself.
Is Masterman Ready? We mean, of course, to face the music at another by-election.
Not “Pygmalion” likely!
After all it is the ordinary old vote of the man in the street, and not the pure and beautiful soul of one the candidates, that really counts at election time.

The second-earliest occurrence of not Pygmalion likely that I have found is from a letter by a person signing themself ‘Disgusted Member’, published in The Milngavie and Bearsden Herald (Milngavie, Dunbartonshire, Scotland) of 5^th June 1914:

SAVING THE EMPIRE BY BILLIARDS?

Sir,—It is really amusing that Mr. Rea of all men should take up his pen on behalf of the Constitutional Club. He thinks that the misconduct of the present Government is a very good reason why we should ignore that misconduct and devote our time to billiards. Opinions like those of Mr. Rea have been the ruin of the Constitutional Club. That’s just what is wrong. The committee says, “Are we going to be bothered fighting against the iniquity, misrepresentation, and treachery of the Liberal Party? Not Pygmalion likely. We would rather play billiards.” Mr. Rea thinks that, because a certain number of Liberals will always support the present Government, it is a sufficient excuse for Unionist apathy. That is why we should pay no attention at all to other people and refrain from making them firm Unionists. That’s just what is wrong with the Unionist Party. There are too many men like Mr. Rea in it.