They most definitely have to obey all traffic signs and signals.
My late father, while cycling, wanted to get to a library which could be quickly accessed by going the 'wrong way' along a one-way street. He thought that he'd be OK to do so as long as he got off his bike and pushed it. However, he was stopped by a police constable who (correctly) reminded him that the law regarded him as still being a cyclist, even when he was pushing his bike. He was told very firmly that he must cycle 'the long way round' to the library and that he'd be prosecuted if he was ever seen contravening the one-way regulations in the future.
Section 36 of the Road Traffic Act 1988, together with Regulation 10(1) of the Traffic Signs Regulations and General Directions 2002 give legal effect to Highway Code Rule 69 (which applies specifically to cyclists): "You MUST obey all traffic signs and traffic light signals".
Chris