From what I remember, the supply shops around Savile Row and Jermyn Saville Row are for traders, which probably doesn't mean you literally have to be in the business, more that you'd have to buy in humungous quantities.
The two fabric shops I know best in London are a bit scattered I'm afraid:
Crescent Trading - off Brick Lane is a proper old fashioned cloth merchants, not massively cheap but beautiful fabrics. I'd check before you go because it was run by two guys and I believe one of them died during the pandemic.
The Cloth Shop at the far end of Portobello Rd (past all the markets). They have a great range of linen. Again check before you go because looking at their website, they may also have moved.
There are loads of others - they're just the ones I've stumbled on.
I've not been to Goldhawk Rd - it sounds like a really interesting place but I think you need to know your stuff better than I do to get the best out of it. I think the Cloth Shop might be reasonably close so might be possible to combine the two.
My view, for what it's worth, is that the current charity shop trend will peak and die down once people realise it's a) very time consuming b) totally random and that, unless you either enjoy the actual hunt (and I do) or have the good fortune to live in a neighbourhood with great shops, then it's not the most effective way of building a wardrobe. I hope I'm right!
I'm afraid I only know about Pimlico because we literally got off the tube once to see what was there.