Thank you, all of you. You're all fab. :)
Update: it's the next-door neighbour's cat. I bumped into her (we haven't spoken before) and I intriduced myself and asked if it was theirs. She said its her daughter's, but she's not looking after it. I said was it ok if we fussed it/fed it occasionally etc, and she said go ahead. She said her daughter got the cat but has neglected it. The daughter also just got a DOG but isn't looking after that very well either!?!
She said the cat had mange, and that they had cream to put on, but the cat just runs away.
We're away next week, but after that I am really tempted to just take the cat under my wing.
I've heard from other neighbours that these neighbours (next door) are quite weird and reclusive. It's just a mum and her daughter. Their garden is FULL of broken greenhouses, old bits of wood, etc, and the shared alleyway between our houses is just full of their old broken furniture, old doors, etc.
I don't even think they'd particularly miss the cat. They have another cat anyway, that looks gloriously sleek and well-fed.
WIBU to look after this little cat? She's only young, barely one. And painfully thin. She's at our back door all the time, and properly sleeps on our laps.