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Of British-English origin, the slang phrase barmy in the crumpet, and its variants, mean: wrong in the head, crazy.

This phrase occurs, for example, in the following from D.C.’s ‘bloody heat’, by Robert Yoakum, published in The Free Lance-Star (Fredericksburg, Virginia, USA) of Friday 26^th August 1983 [page 14, column 5]:

Never thought I’d miss English weather, but at least I now understand why my colleagues from London consider Washington a hardship post. […]
[…]
This heat would go a long way toward explaining the idiosyncratic behavior of Washington politicians. Anyone whose cranium is baked for several weeks on end must become a bit barmy on the crumpet.

In the phrase barmy in the crumpet, the noun crumpet designates the head, and the adjective barmy means: crazy.

—Note: The adjective balmy, when used to mean crazy, is an alteration of the homophonic adjective barmy.—Cf. origin of the adjective ‘barmy’.

These are, in chronological order, the earliest occurrences of the phrase barmy in the crumpet and variants that I have found:

1-: From Chuckers-out, published in The Waipawa Mail, Waipukurau, Kaikora, and Hampden Advertiser (Waipawa, Hawke’s Bay, New Zealand) of Tuesday 30^th January 1883 [page 4, column 3]—reprinted from the London slang magazine The Rag, or Weekly Scorcher:

Chuckers-out are of two blooming sorts generally—simple and compound. The simple chucker-out is sometimes a bit barmy in the crumpet, and is only kept for the sake of show, and to prevent the sweet tarts behind the bar hollering out when a shindy starts. Such is the chucker-out you’ll find in quiet sort of pubs which only toffs use—toffs that never have the ghost of the slosher in them.

2-: From Sporting Notes, published in The Sporting Times. Otherwise known as The “Pink ’Un.” (London, England) of Saturday 15^th May 1886 [page 1, column 1]:

He had been showing signs of mental aberration for some time, and when his Tart [cf. footnote 1] called at the office it was necessary to explain how things were with him. Master delicately hinted at a little weakness of intellect, doubtless due to overwork. Blobbs mildly suggested that too much study had temporarily upset him. Shifter frequently felt like that himself of a morning, and there was really no call for the alarm. And the poor woman would have gone away utterly ignorant of the real state of the case, had not the Talepitcher, in his refined and dainty style, observed:—
“You see, marm, that your okeblo [cf. footnote 2] is a bit balmy in his crumpet.”

3-: From On the Use of Slang, by ‘Pitcher’, published in The Sporting Times. Otherwise known as The “Pink ’Un.” (London, England) of Saturday 31^st May 1890 [page 3, column 1]:

The sphere of the common or garden human brain box is ever widening, and of late years has been extending at a prodigious rate—possibly in no branch more than in the Volapuk of Vhitecha-pel. Only the balmy in the crumpet take “all knowledge to be their province.”

4-: From Slang and its Analogues Past and Present. A Dictionary, Historical and Comparative, of the Heterodox Speech of all Classes of Society for more than Three Hundred Years ([London]: Printed for subscribers only, 1891), by the British lexicographer John Stephen Farmer (1854-1916) and the British author William Ernest Henley (1849-1903) [Volume 2.—C. to Fizzle. s.v. Crumpet, page 224, column 1]:

Balmy in one’s crumpet.—See Balmy, sense 2 [cf. footnote 3], and the foregoing.

Notes:

1 Here, one’s tart designates one’s wife or girlfriend.—Cf. origin of ‘tart’ (promiscuous woman).

2 The slang noun okeblo means: a bloke. The Irish Republican activist Michael Davitt (1846-1906) explained the origin of this slang noun in the following from Leaves from a Prison Diary; Or, Lectures to a “Solitary” Audience (London: Chapman and Hall, Limited, 1885) [Lecture 16, page 109]:

Thieves’ Latin.—[…] Its chief peculiarity consists in reversing the position of the syllables of a word containing more than one syllable, and making two syllables of all words having only one in ordinary pronunciation by adding a vowel or liquid consonant to the first or second part of such word. […]
[…]
As some words will not admit very well of the necessary transposition of syllables needed to disguise the talk from listening victims or enemies, the first syllables of such words, if immediately following each other, will change places, so that the first syllable, letter, or letters of the second word will become that or those of the first word, and vice-versâ. For instance, if Jack had made the discovery that a person whom himself and Bill were following had only a silver watch, the disgusting fact would be told to Jack as follows:—
“I jay, Sack, the okeblo’s wack’s clite;” which in slang would be—“I say, Jack, the bloke’s clock is only a white one;” and in English—“The fellow’s watch is only silver.”
The letters “J” and “s” of the words “Jack” and “say” are exchanged; the ordinary lingo rule is followed in reference to the word “bloke” and the “cl” of the word “clock,” and “w” of “white” are exchanged as in the case of the letters “J” and “s.”

3 “Balmy, sense 2” is as follows in Slang and its Analogues Past and Present. [&c.] ([London]: Printed for subscribers only, 1890), by John Stephen Farmer [Volume 1.—A. to Bys. s.v. Balmy, page 111, column 1]:

Dull-witted; thick-skulled. In this sense balmy is used up and down the whole gamut of imbecility from mere stolidity to downright insanity. Popularly used, it signifies in most cases little more than shallow-brained or muddle-headed; or, to use slang equivalents in their most familiar sense, ‘to be touched,’ ‘to be wrong in the upper story,’ ‘dotty.’