anywhere in the UK that you can train up in and then hire an old warbird without a significant up front outlay ... within a reasonable driving distance
You don't want much, do you?
Frankly, I think that you're going to have a difficult time finding such a situation.
Skysport (near London) will rent you an ex-RAF Bulldog. If you have your heart set on something more exotic, my suggestion would be to: (i) keep your eyes and ears open for likely situations, and (ii) in the interim, build experience on whatever relatively complex / high performance airplanes may be available to you (especially tailwheel).
There are a few syndicates flying around, but for the number of hours I would be looking at, I'm not really sure that they are my cup of tea (£10k buys me an awful lot of hours on other types!).
You can't make an omelette without breaking some eggs.
You might consider buying a syndicate share, doing whatever flying you want to do, and then selling your share. If you pick an airplane in good condition, you shouldn't lose much money, if any.