Green’s Dictionary of Slang

1. a hangover.

1906		Sun. Times (Perth) 11 Mar. 1/1: One scribbler speaks contemptuously of the drink discussions [and] blames the conference for one fat head.
1914		‘Bartimeus’ ‘The Night Watches’ in Naval Occasions 179: Now put the cups back, and come and show me Arcturus – if you have shaken off your fat head!
1929		(con. 1916) F. Manning Her Privates We (1986) 112: Was I drunk last night? [...] I’ve got a bit of a fat ’ead.
1936		J. Curtis Gilt Kid 86: I’ll let you have something to cure your fat head.

2. a headache.

1917

‘Taffrail’ Sub 219: Our cabins [...] were distinctly frowsty. [...] I used to wake up with every symptom of what is generally known as a ‘fat head.’.