Maybe that's specific to UK? I've never had to upload ID at any of the general domain registrars but am not in the UK. Maybe you're looking for a registrar that doesn't do that for UK residents, specifically. (or you might need to use a VPN or something, dunno)

Privacy aside most people tend to buy a domain at the cheapest registrar (keeping in mind whether to renew at that registrar later or transfer out to a new one when the time comes). There are domain registrar sites people use for comparison, been a bit since I've needed to shop around but sites like https://tld-list.com/ can be helpful. You could try doing a lookup there and work your way down the list until you find one you like.

Or if you're really set on looking for a service that actually advertises no KYC then looking through this list may help https://kycnot.me/?categories=domains (njalla is one of the listed ones there as someone else mentioned).

Can you use a dummy address or a friend's address in another country and/or sign up via a VPN to mask your location? There are companies that will open and scan your mail and make it available online so it could be a real address. But they are a little pricey. It's likely due to UK laws not the provider themselves, so probably all reputable providers (that follow the law) may require it.