In British English, the noun crumpet is used to denote:
– women regarded collectively as objects of sexual desire;
– sexual intercourse.

Therefore, a bit, or a piece, of crumpet denotes:
– a woman regarded as an object of sexual desire;
– a sexual act.

—Cf. also:
– the noun crackling, used in British English to denote women regarded collectively as objects of sexual desire—and the phrase a bit, or a piece, of crackling, denoting a woman regarded as an object of sexual desire.
– a crude phrase: ‘to see a woman’s breakfast’.

The earliest instances that I have found of crumpet used in a sexual sense are from The Gilt Kid (Jonathan Cape – London, 1936), a novel set in London, particularly in the underworld of Soho, by the British author James Curtis (born Geoffrey Basil Maiden – 1907-77). In the following passage, crumpet means sexual intercourse—Context: The Gilt Kid has just seen the woman he loves, Maisie, in the company of her sugar daddy, Bedborough (whom the Gilt Kid nicknames ‘old Bedbug’):

A coffee-stall, a red oasis of light and warmth, was before him. It was not such a bad idea. He went up to it and laid a forearm on the greasy counter, resting his left foot on the wheel-spokes as though they were the brass foot-rails of a bar. The man behind the counter had ginger hair. He rose, glad to have a customer to break the monotony.
‘What d’you want?’
‘Tea please, and […] a sav and a slice.’
‘Cup o’ tea, sav and a slice,’ intoned Ginger. […]
‘You look miserable, mister,’ said Ginger taking the sixpence. ‘What’s a matter? Ain’t you had no crumpet?’
‘No.’ The monosyllable was curt. It was too bloody true that he’d had no crumpet.
‘Blimey,’ Ginger’s back was turned. He was ringing up the sixpence on the cash register. ‘Fancy staying up as late as this and not having no crumpet. You must be a bit slow, mister. No wonder you’re miserable.’
Another man came up to the stall.
‘Cup of tea and hot pie,’ he said.
‘No,’ repeated the Gilt Kid, ‘I didn’t get no crumpet.’
The other customer looked at him curiously. The Gilt Kid was oblivious. […] Ginger was right. He hadn’t had no crumpet. He took off his hat. There were plenty of cows up round Victoria Station.
‘No,’ he said to Ginger. ‘You’re right, mate. I didn’t have no crumpet. She went off with old Bedbug.’
He drifted up back to Victoria again.
The other customer looked at his retreating back significantly.
‘What’s the matter with that bloke? Doolally?’
‘No,’ said Ginger, sympathetically, ‘his tart’s got the bedbugs so he didn’t get no crumpet. Enough to make a bloke a bit miserable staying up all this time and not getting none.’

James Curtis used the word again in Look Long Upon a Monkey (Arthur Barker – London, 1956), as mentioned in the review of this novel by Elizabeth Harvey, published in The Birmingham Post (Birmingham, Warwickshire, England) of Tuesday 30^th October 1956—to lumber means to engage in sexual play with, to copulate with:

Quicks wits and a benevolent frame of mind are needed for full enjoyment of the tense drama unfolded in Look Long Upon a Monkey, the benevolence for overlooking the impossible brassy, primitive and even fictionally outmoded women characters, and the wits for learning the unfamiliar language, and interpreting such phrases as “lumber as much crumpet as he fancied” and “not if she was a dirty grass.” For this is how the three escaped prisoners talk who are making themselves at home in the kitchen, when a just-married couple and their semi-sponging friend Tony return home to their lonely house on the Wolds after a highly successful day at the races. Theories which the trio had aired before on the problem of criminals can now be put into practice or readjusted.

In the following passages from Borstal Boy (Hutchinson & Co. Ltd – London, 1958), an autobiographical novel set in a borstal by the Irish poet, short story writer, novelist and playwright Brendan Behan, bit of crumpet denotes a sexual act:

‘You can smoke your head off in here if you got any snout or the money to buy it. You can buy scoff [= food], your breakfast, dinner and tea, chocolate. Except drink or a bit of crumpet, I suppose.’
[…]
‘’Course you can talk in Borstal. Talk your bleeding ’ead off if you want.’ Except, he said, for the Silent Hour, when you had to sit in the dining hall reading a book or writing a letter, or else doing nothing but be quiet. Littlewood laughed and said that even at this period you could shove your head in your hands and think about the last bit of crumpet you had outside. That was, he added, if any us pink arses had ever had a bit, which he doubted.
[…]
‘Well,’ said Joe, ‘’ow about your old man and woman? I reckon they’re over forty.’
‘My Mum’s thirty-seven,’ said Charlie, ‘she was twenty when I was born, and my old man’s a year or two older.’
‘And you don’t think they’re ’aving a bit of crumpet still?’ asked Joe.
‘You shut your bloody mouth,’ said Charlie, in a bursting rage, his lips set stern, and his fists clenched.

In the following dialogue from A Taste of Honey (Methuen & Co. Ltd – London, 1959), a stage play by the English dramatist and screenwriter Shelagh Delaney (1938-2011), bit of crumpet denotes a woman regarded as an object of sexual desire—Jo’s mother, Helen, evokes Peter, who has left her for a younger woman:

Jo: What happened?
Helen: He’s gone off with his bit of crumpet. Still, it was good while it lasted.

The Daily Mirror (London, England) of Wednesday 5^th February 1964 associated the figurative use of crumpet to Army slang—Steptoe and Son was a British sitcom about a father-and-son rag-and-bone business; 8½ is the title of a 1963 motion picture by the Italian film director Federico Fellini (1920-93):

MORE PROTESTS OVER STEPTOE

An old Army phrase in last night’s “Steptoe and Son” TV programme started the phone bells ringing in protest at the BBC. Steptoe senior wanted to see “Nudes of 1964.” Harold objected.
Finally they decided on the Continental film “8½.” As they were leaving home, Harold asked his father if he had taken his glasses.
Retorted Steptoe senior: “I don’t want my glasses. I’m not going to see any crumpet.”  At once, angry viewers started ringing.
A BBC official said: “We have had some viewer reaction, but we can’t go into detail.”

The Daily Mirror evoked the incident again the following day, Thursday 6^th February 1964:

You can’t please all the people all the time.
The Daily Mirror learned this years ago; the BBC are still learning.
“We have had some viewer reaction (they mean complaints), but we can’t go into detail,” said their spokesman about a programme this week.
Somebody, somewhere—or a lot of people nowhere—had telephoned BBC boss Mr. Hugh Carleton Greene to protest that Steptoe Senior had said something to his son about “crumpet.”
Of course, Steptoe Senior might equally well have referred to the ladies as “popsies” or “birds” or “bints” or “dolls” or “dames.” Or even as ladies.
But who wouldn’t prefer to be called “crumpet” rather than “strumpet”? It’s a matter of taste.

On several occasions in the 1960s, the Church of England priest Geoffrey Bryan Bentley (1909-96) used the word crumpet in the sense of sexual intercourse. The following is from ‘Church will not stop pagan sex by teenagers’, published in the Evening Express (Aberdeen, Aberdeenshire, Scotland) of Thursday 14^th May 1964:

An appeal to church people not to be shocked by modern trends in sex life, particularly among the young, is contained in “God and Venus,” a pamphlet published by the Mothers’ Union.
The author, Canon Geoffrey B. Bentley, of Windsor—father of four children—calls for understanding of modern attitudes towards sex.
“The sooner it is recognised on all sides that it is not within the competence of the Church to stop pagan teenagers having sex together, the better for everyone concerned,” he says.
[…]
“If they think we are saying that sleeping alone is intrinsically preferable to a piece of crumpet, or that virginity is in itself superior to the process of losing it, they will merely conclude that our heads need examining . . .”

Geoffrey Bryan Bentley used the word again two years later, as reported in The Queen’s parson startles the viewers, published in the Daily Mirror (London, England) of Tuesday 18^th October 1966:

Hundreds of TV viewers were shocked last night by a phrase used by the canon of a Royal Chapel when he was discussing a controversial Church report on sex.
Canon Brian [sic] Bentley, of St. George’s Chapel, Windsor Castle, was talking on the BBC programme “Twenty-Four Hours.”
His subject: The shock report “Sex And Morality,” commissioned by the British Council of Churches—a report which refuses to condemn sex outside marriage as always wrong.
Canon Bentley said: “A bloke may be in the habit of having a nice bit of crumpet on Saturday night.
“But if he wants to join the Church as a Christian he will have to give this up.”
A BBC spokesman said: “The canon’s comment caused a number of people to telephone us.”
Canon Bentley said after the programme: “Crumpet is just one of the words one hears about having sexual intercourse.
“That’s why I used it. I try to use the language of the people.”

This led Iain Hamilton to associate crumpet in the sense of sexual intercourse to “trendy Anglicanism” in The Illustrated London News (London, England) of Saturday 12^th November 1966:

Success was always a suspect word to those who could see the skull behind the grin and the feet of clay behind the soft leather, and who knew the quaint malice of the bitch goddess. But that was in the days when lip-service at least was paid to values which tended to inhibit slightly the instant attainment and open enjoyment of power, wealth, and, in the phrase of trendy anglicanism [sic], unlimited crumpet.