I asked a Royal Enfield friend, and he can recognise bits of RE amongst the cycle parts. He also suggested that the motor looked "a bit AJS", and after a scrounge I came up with the attached image: the AJS 996cc "S2" twin new for the 1931 season. It must have lasted only a year or two because by 1933 I think AJS twins used the familiar Matchless motor with the round timing chest and square tappet covers. I've never seen one in the flesh.
Anyway I'm pleased it's AJS, because I had feared that the motor might have come from an (Australian) Howard cultivator - a number of unfortunate specials have been built around this motor.
Apparently the bike is owned by Shaun Jackson of Portsmouth, and goes by the name of "1929 Jackson Indomitable 1000". Given the motor and gearbox, "1929" is an interesting date - perhaps the gas lights take it back a few years.
By the way, how do you register such specials in the UK? We have a historic registration scheme over here that doesn't cater for specials.
Leon