Green’s Dictionary of Slang

Birmingham screwdriver n.

also Brummagem screwdriver
[the supposed oafishness of the Birmingham worker who would rather hammer in a screw than use the correct tool; despite normal racial stereotypes (and their supposed jobs) a US usage yiddish screwdriver has been noted c.1939]

a hammer.

[UK]Partridge DSUE (8th edn) 142/1: Brummagem screwdriver A hammer.
ADC Theatre (Cambridge, UK) Theatre Lex. 🌐 Birmingham Screw n. A malformed wood screw, lacking a slot in its head and so ideally suited to insertion with a hammer ( a.k.a. Birmingham Screwdriver).
Variable Pig mag. May 🌐 After a few efforts to remove the old units subtly, I resorted to a 2kg Birmingham screwdriver and bashed the hell out of the units.
[UK]J. Meades Empty Wigs (t/s) 768: ‘Talk about bodging. Him an’ his Birmingham fuckin’ screwdriver. Never heard of rawlplugs!’.