Suffering pet , what to do?

I know it isn't what you want to hear but I believe you have to speak to your neighbours. Are you sure they haven't already taken their cat to the vet, or that they are deliberately neglecting it. Old cats often look dreadful, skinny things. If it were my cat, I'm sure I'd be making that final loving decision but it isn't my cat, it belongs to your neighbours.

There are a few awful cat diseases that could be causing the poor old things problems
I'm sure its actually illegal not to seek medical treatment for a poorly pet.
I suppose you could dress up a policeman?
Sorry for making light, because its a horrible situation.

Do you think that the owners would care if the cat ' dissapeared'. It doesn't sound like it to me. In fact it's downright bloody cruel. If it were me I would take it to the vet to be humanely put out of its misery.

Have you thought of contacting the Cats Protection League? www.cats.org.uk/what-we-do/about-us/helpline
A friend has done a lot of work with them and they have a helpline which may be able to give you advice.
I know they work with vets to alleviate suffering in cats. Tricky problem for you.

I think the vet would examine the cat whether it belonged to you or not. You do need to be careful though. My aunt was sorry for a neighbour's neglected cat which kept coming in, she took him to a vet and paid for his treatment, and was then accused of luring him in and stealing him! This wasn't a valuable pedigree cat but a mixed moggy. (Eventually not only that one but also the mother cat decided to move in with her)

Strangely enough, I've been in the exact situation you find yourself in
What I did doesn't mean I'm suggesting it was right, but well, I did it
I took my neighbours cat, who very obviously had a badly injured leg, which hadn't had medical attention, to a cat sanctuary, and told them she was a stray.
I never told a soul, and feel I did the right thing.
My neighbours told me the cat was limping "for attention". They had various pets, which they got rid of as and when the novelty wore off, or neglected. They really weren't animals lovers at at..