Green’s Dictionary of Slang

(UK Und.) to put forward, usu. to offer a name to the police.

1928		C. McKay Home to Harlem 46: Let the boss-men stick them jobs up.
1935		Thieves Slang ms list from District Police Training Centre, Ryton-on-Dunsmore, Warwicks 10: Stick it up: Put up [a] job or give information to thief or police.
1956		‘Charles Raven’ Und. Nights 166: As soon as he was satisfied that I knew how to graft and was a fairly steady reliable type, he began sticking up jobs to me.
1974		G.F. Newman You Flash Bastard 31: Be a long job checking just names through CRO – that’s assuming whoever stuck up the right name and has form.