does the Apple Developer programme support sole traders who are VAT registered in the UK?
Yes, but with one proviso: you cannot use a "doing-business-as" pseudonym. I.e. if your name is actually John Smith but you like to refer to your business activity as Cromulent Enterprises, then tough; Apple will refer to you as John Smith throughout the app store.
I was a (UK) sole trader when I first started selling apps, and I actually started a limited company in large part in order to get a more corporate name in app store listings. Of course as soon as I did that, Apple de-emphasised the business name in the app store listings; previously they were shown more prominently alongside app names.
Being a one-person limited company involves a bit of bureaucracy, i.e. PAYE - it's more work than being VAT-registered, for example - and you have to pay employer's National Insurance, unless you have a second employee. Think twice (get advice) before doing it.
DUNS numbers are not required for non-company enrolment.
Note that in the US, they have a concept called "disregarded entities" - as I understand it, you can create a company which is sufficiently real that it gets a DUNS number and can register with Apple as a business, but it is "disregarded" for tax purposes, i.e. you are a sole trader as far as tax is concerned. It may be that this is the reason why UK sole traders are not handled quite as you might like.
VAT registration or not doesn't change anything.
(Disclaimer: I've been a limited company since 2018, things could have changed.)