I used to brew my own 30 or so years ago, and also tried my hand at cider making from my apple tree and 'wine' from some other fruits. It's quite good fun, and there is something magical about the transformation of a container of sludgy, bubbling liquid into something crystal clear.
'Two Brewers' off license in Forest Hill was then one of the best known home brewing sundries shop, owned by the people who set up Provender. There must be other suppliers still in business, and they should be able to advise, and also supply equipment, sterilising materials and yeast,
I suspect that you needn't be in for too much outlay to start with. We planted a couple of vines that didn't do as well as yours seem to have done - blame global warming - so it didn't seem worth spending money on expensive kit. However, the only really expensive item seemed to be the press. Fermentation and storage vessels shouldn't break the bank. I wonder whether it wouldn't be worth trying a dry (or should I say 'test') run using a kitchen juice extractor to see what results you might get, and if your first results seem at least redeemable with practice, buying the press only then.
I received a fair amount of advice, but will only pass on the fairly obvious: 'remove your socks before treading the grapes'.