Was she allowed to do this?

Technically, yes, but if the cat was staying in your house as a pet, (and you'd taken him on as a pet including a collar and such) then I'd say no. If he was a stray and neither your family or neighbours were taking him on as a proper pet, then I'd say that he would have a better life if someone adopted him from a rescue centre. If you and your family want a pet cat, then you should go to the shelter, sign the relevant papers and get him back.

Nope Wood Green isn't a kill centre :-) http://www.woodgreen.org.uk/

Nothing you can do. It was a stray

Feeding a cat does not make it yours.

The neighbor had every right to take astray cat tot he rescue since no one owned the cat. No one payed any vet bills, no one chipped the cat so it belonged to no one.

You can be upset all you want you didn't own it.

ring the rescue centre and explain, or better yet call out there. Im sure there would be no problem getting your cat back. Unless its already chipped and then it belongs to someone else. If you do get your cat back get it chipped immediately and neutered, it can help prevent 'wandering'. Do it fast because they might have your cat put down. Your neighbour was inconsiderate. I hope you get your cat back.

You do nothing but if you go to the shelter to adopt, you plan on keeping this cat INSIDE if you want to keep it as a pet. No excuses. The cat was outside all the time so it really was no one's cat.

The neighbor could have talked to you guys first before taking it to the shelter.

Unfortunately, she should have asked you guys first and given you a choice of keeping the pet. Not much you can do unless pet is still at the rescue cte.

You could explain to the rescue centre what hapened and they may give it back to you or you could officially adopt him!

Nothing you can do. It's done. Get over it. Move on.