You might want to consider the evolution of the British motorcycle through the 1930s. Often the late 1930s model was very similar to the early 1930s model from the same maker, but carrying tens of extra pounds.
As 33d6 says, go with what tickles your fancy. As well as the usuals (Triumph, BSA, Norton...), have a look at Ariel, Rudge, Royal Enfield or even the smaller makers. I'm not much of a 1930s person (a bit modern for me) but some very nice models came out as the depression eased in 1932-3.
Good luck!
Leon