In all cases above, I would replace the noun "threat" with the verb "threaten."
The neighbours threatened to go to the police.
The neighbours have threatened to go to the police.
Your 1st & 2nd examples are grammatically correct, but awkward in their use of the present-tense verb "express" which sounds as if you're narrating the scene in real-time. If that's your intention, they make sense but would still sound more natural with "threaten" instead.
The neighbours threaten to go to the police (right now, as I'm narrating events to you).
Your 3rd example also mixes verb tense in a way that doesn't sound natural to a native speaker. You're expressing a simple present-tense threat about a present-tense action. It most likely should be a past-tense or present-continuous threat, and a future-tense or present-continuous action.
As written, I would say it is not grammatically correct unless the neighbours routinely state their intention to go to the police for all sorts of things. It's like a detached narrator watching from the outside & saying "The neighbours announce that they enjoy skiing." Ok ... so what?
Some further examples:
The neighbors express a threat that they are going to the police.
Happening right now. They're going to the police as I narrate it.
The neighbors expressed a threat that they will go to the police.
They expressed it an hour ago & will go at some future time. This would be most natural if you're explaining completed events to somebody else, and the neighbors haven't yet gone to the police.
The neighbors are expressing a threat that they will go to the police.
Happening right now as I explain it, they'll go to the police in the future.
This would be most natural if the neighbors are still arguing right now & you're explaining what's happening.
Would a native English speaker understand all of the examples regardless? Probably. If that's all you care about, then say any of the 3.
That said, if you're interested expressing yourself in a way that sounds normal to native speakers and clearly matches your intended meaning, then hopefully these explanations clarify things a bit.