I'm fairly certain that we sold some Sunbeam Rapiers with a white band in their tyre walls in the late fifties and we used to fit add on white walls (we knocked the outside tyre bead off the rim and tucked these white rubber circles behind then reinflated the tyre to trap them.)
The last production car I remember with white walls was the Austin Westminster A105.
I think the Austin A105 was last built in about 1957-58, mind I could be wrong, but I'm pretty sure that Vauxhall Cresta's, and possibly Ford Zodiacs mk2 ,built until about 1962, were fitted with white-walls as standard, Mal.