It would be worth trying to put it together with largely 1950s bits, and it needn't cost much if you go to a cycle jumble and steer clear of exotica.
It is, I think, a road/path frame, as it has brake drillings, so it would probably have been built using GB bars and stem , Williams C34 chainset, GB side pull brakes (but Weinmann were also available then), and a Brooks saddle. Early Brooks saddles can be costly, but the other bits are cheap enough and readily available.
As for wheels, you could go for some 70s or 80s small flange wheels, either 27s or 700c. No one will notice they are later as long as you fit some Amber wall tyres. Or go for sprints and tubs, which are often to be found at bargain prices at jumbles.
Good luck with the build!