I think that the RSPCA is the only way to go here... there is no-one else that fills that position, shit as they can be IMHO. They have no more power in law than you or I despite their uniforms and titles but rely purely on civil law and the assistance of the police in criminal cases just as we have to, the difference being that they have the money, police contacts and access to lawyers to bring prosecutions.
I have found that what they often need is a kick up the arse to act and that means lots of pressure. Frequently it helps to have several complainants to get them to move fast.
You might either like to give details of the dog (health, breed, sex, colour etc and FULL address, including postcode, with owner's name if you know it) to a few determined and articulate friends/family and ask them to report it to. If that's not possible and you're happy to do so please pm me with the details and I and some rescue friends will make those calls along with you. For your peace of mind I'll happily give you my phone number to chat to me and a list of people to contact in rescue so you can check out who I am and that I'm genuine as I quite appreciate that you might not want to give your neighbour's details to a stranger on the net without ensuring that they aren't who they say they are.
THE PROBLEM WITH THIS IS that the poor dog will quite probably end up put to sleep as a result. It will depend on many factors.... the condition/cost of treatment/chance of recovery, the age of the dog, the breed and the dog's temperement.
Another possible solution is to persuade the neighbour to approach a strictly no kill rescue and ask them to take on the dog and appeal for help with vet fees. That's going to be bloody difficult. I can certainly help a sympathetic rescue, if they are willing to take the dog, by appealing for help with funding an operation but I can make no promises. Depends where you are... I MAY be able to identify possible rescues but unless the owner is willing to sign the dog over to one we're flogging a dead horse. No-one is likely to pay for the op to leave the dog with the owner, who either doesn't care or who doesn't have the courage/sense to do the right thing.
Do I take it that the RSPCA etc can't treat the dog because the owner isn't on benefits? Because if so it MIGHT be worth finding someone understanding who is and cheating a bit by taking the dog on as their own and then taking him to their local RSPCA surgery and getting the op done as a benefit-claiming and thus eligible owner... and then getting the dog into rescue.