Hi Chrstine, I do know how you feel as I has this problem with my son at the same age, although he is now nearly 7! First and foremost, don't worry about those neighbours, get yourself and your son sorted. I doubt it bothers them that much anyway you are being sensitive to it I expect because you are tired and fed up with everything not working for you!
So long as you are confident he is not starving hungry I would guess it is attention he seeks, and of course he keeps getting it! If you do feel you have to give anything at all, just cooled boiled plain old water in a bottle, just in case he is thirsty because if he has milk, that's great isn't it and he will love that. If he only gets water, he may think well, maybe it's not such a great deal waking up and making a fuss afterall. I tired the same technique as the other respondant and it worked. Don't ignore him, but don't stay too long. Just go in and talk quietly, let him know you are there, but be firm. STay a few mins, then reduce the time you stay each time you have to go in. LEave him to cry a little in between though, don't just run back in as soon as he starts crying! I also tried placing a chair near his bed and sitting on it with him, not talking, just read my book and he went back to sleep. Then I moved the chair nearer the door each night, and eventually sat outside his door, he knew I was there. He eventually [about 2 weeks] developed confidence to go back to sleep on his own again.
Wishing you a restful night, and hoping you get this sorted soon.
Take care,
S.