I have a question about using the word "neighbor'

I think in this context, neighbors would be fine, as it's obvious they're not referring to his new neighbors because if they knew who his new neighbors were they would know where he was. People sometimes do say "old neighbors" or "former neighbors" when it isn't clear the person is not talking about the current ones, eg. "One of my old neighbors used to throw his dog's poop over the fence into my yard." If someone said that without the use of the word "old" one could reasonably assume it was one of the current neighbors that used to do it. That's not to say that native speakers don't say things ambiguously like that all the time, though. That said, if this is a question on an ESL test they may expect you to specify anyway.

Old neighbours works. Former neighbours is OK too, if a little bit formal. I've often heard people say "ex neighbours" too.

I would say “old neighbors” almost exclusively here. “Former” is fine but other commenters are right, it’s pretty formal. I suppose you could derive who you’re talking about from context, but to avoid confusion I would say, “old neighbors.”

Neighbor is simultaneously a relation and a status, and it isn’t always clear which. This is more an issue of logical elision, the speaker doesn’t specify which neighbors, old or new, because it’s assumed the listener knows the new neighbors couldn’t have the address. Anyone doing chatbot research before the current AI boom can tell you, humans are always making a set of assumptions about what their listener knows, it’s a big part of verbal communication.

It would be entirely clear what E means, based on the most cursory pragmatics.

So from a purely grammatical stance, the example you gave would generally be understood. You could say "old neighbors" if you wanted to be more clear. That said, in the situation you provided, if these people are close enough that they would be keeping in touch after a move, you'd probably just refer to them by their names rather than as "his neighbors". At least in the region of the US I live in, being neighbors doesn't automatically mean you're friends or would keep in contact with each other.

I think that I would be the same with the word neighbor in your native language