When I ask my partner who comes from the North of England a question (like did you empty the bins), he answers often with this phrase 'I will have done'. I have no idea what this means. Can anyone illuminate me? Also, is this grammatically correct, or not

It's either, ''I will get it done'' or ''I probably have but I can't really remember''. I'm from Northern England, and it could be used in either context.

I'd say 'I will have done,' is 'bout tha same as 'I'll git it done,' which is what a pardner from the south of America mite say if asked if he had been asked "Didja empty the trash can?" as a bin is a place to store things you're gonna use and not toss out. 'Cept for recyclables which for some reason are usually kept in bins.
"Also, is this grammatically correct, or not" No, I don't think so, you left off the question mark, so it is not a complete question. Also the "or not," while correct, is superflu..superful..sup vre..not needed, as it is implied in the question. But that is a matter of style not grammar.
As for 'I will have done;' it is as correct as your reply of "Do it now!" The subject of that is implied " (You) do it..." A participle or particle or whatever it is called is implied; "I will have done (it by noon.)"
Leavin' words and/or entire phrases out of sentences is acceptable, if not quite 'correct,' in informal speech.
I think. I don' wanta imply that I am and an expert on grammar or style, I mean there are some Grammars with style, others still dress like they did while in their twenties. I don't care one way or another, as long as they don't nag at me all the time to empty the container of discardables.

Thanks for the laughter. It sounds very funny so I think your partner is trying to make you laugh. i.e. its a joke.

Louise Marilyn Smith was, for at least 2165 days, more than happy to have ME for a boyfriend, despite her belief that I could be A TRIFLE REPETITIVE at times.

It is a short cut for "I will have it done".

I would understand it as "I haven't done it yet, but I am going to do it".

He is a Northerner...Who knows what it means....probably not even him...