If you have wire wheels, then you'll need to find someone who is capable of balancing them. This normally involves using a set of cones to centre the hub on the spindle of a conventional tyre balancing machine. Your best bet for this is a well-established tyre specialist rather than a Kwik-Fit type place.
It's quite common these days to order tyres online and have them delivered to a local tyre place for fitting and balancing. I'd ring a few places and see what they say. Having a quick google, I'd start with
https://lloyd-tyres.co.uk/ as they've been in business since 1967 so will probably have the necessary kit.
