Ok, so first of all, is the uniform something outlandish?
If he needs green trousers & a green/orange checked shirt & a green badged blazer, then the school should be at least considering letting y11s attend in 'smart casual attire'. They could spin it as preparation for dressing appropriately for sixth form. It would be a pragmatic solution to the fact that loads of families will be in the same boat, & foster goodwill. It should not be beyond the ability of a competent SLT to devise a dress code & enforce it (schools with sixth forms manage fine).
If OTOH it's black/grey trousers & plain white shirts, plus a school jumper, then I don't think that's an unreasonable ask.
A boy in his late teens really should have access to an interview/funeral combo - buy them on the generous side obviously if he's still growing! In fact Asda used to sell perfectly acceptable machine washable suits for about £30 - one of my teaching colleagues used to pride himself on the fact that his entire working wardrobe consisted of two of them on weekly rotation...he wasn't a fashion plate, but he was completely presentable.
For anything logo'd, I think you could reasonably contact school & see if they've got a suitable jumper or whatever in lost property. If they don't, I think you could further push for a plain item in the appropriate colour.
I get it's bloody infuriating. I'd literally replaced my ds's entire uniform this time last year to see y11 out, & then he never wore it & it all went to charity.
But I think it'll be easier if you look for compromise solutions rather than just refusing. There are benefits to uniform - it's a well trodden argument on both sides, but the vast majority of schools do have a uniform, & they're honestly not just doing it to piss off parents. You have my full sympathy though - the buggers grow like weeds at this age.