Do NOT send him to the RSPCA. I'm also from the UK and have volunteered at a RSPCA rescue center. More often than not the birds are themselves neglected at rescue centers, 2 toys (if any!) and put on seed diets in horrid cages etc. Also, they more often than not won't take on animals that haven't been found as a stray/dumped, if you try and 'take' them to the RSPCA, they usually will decline to take the bird.
I found a place in Essex that are a boarding facility but also double up as a rescue who help rehome animals too.
Small Animals Boarding - Pet Boarding and Pet Hotel in Colchester, Essex and London
(Found from here: Preloved | a non-profit boarding rescue rehome and sanctuary for pets other in Ardleigh, Colchester, Essex)
Doesn't say anything about being a rescue on the website, but if you ring the number (I tried to get more information, went to answerphone and I HATE leaving messages!) they mention that they're a rescue too (and obviously the preloved advertisement). I think if you explain that you found the bird where you did, in the condition you did, vet checked it etc and want to rehome if they'll probably take it on.
You could also go on websites such as preloved/gumtree etc. Advertise that you have a male moustached parrot who has been vet checked etc but comes with no cage etc. See if you generate any interest, meet the people interested and decide on a home for him.
I found a lovely home for my rescue cockatoo on preloved, a man had lost his macaw 2-3 years before and felt ready to take on another bird. We got in touch, arranged a meet after discussing in depth his situation (took the cockatoo with us as it was a looong journey) and then handed him over. The guy was genuine, got to see his home, the cage setup, etc.
Other than that there is 'birdline' which has mixed reviews. I think you have to pay a small fee to give up your bird though. They'll find the bird a foster home until they can go to their permanent home, but are still 'owned' by birdline.
You could always look for breeders/people with aviaries etc looking to expand their flock. I rehomed 4 of my budgies to a man who was looking for more budgies to fill up his aviary, they had heated sleeping areas, large aviary area, lots of different activities they could do and nest boxes for those that were interested. It was really lovely, went and checked it out and agreed to give them to him and then brought the budgies to him a few days later.
If you need help writing an advertisement or vetting people, feel free to ask questions. If anyone else can come up with any other idea for getting this parrot somewhere safe then suggest away!
Just as a little reminder in case you think you may profit from this, you most probably wont so I wouldn't really try 'selling' him, but do not advertise as FREE as you'll get all sorts of people trying who cannot/will not care for him and will just try and sell him on to someone else. 'Small asking price' or 'Price can be discussed upon meeting' etc will put off some people looking for an easy profit.
