I think this could be Headless, a Channel 5 film from 2000.
Probably not British, but been trying to track this one down since the 80s when I saw this as a kid on TV.
All I remember is it was in some kind of block of flats or something, and there was a big mainframe/minicomputer sized computer that looked like a washing machine (or maybe it was a washing machine), that moved around a room. There was a waste disposal unit involved in the murder, or disposal of a body, and there was a guy in a lift I think who was standing next to another guy and spotted the first guy had a paper bag with a head in it.
Freaked me out at the time, but probably rubbish. I've not really found much about it though.
I did ask before on another forum and a suggestion came up of Crosstalk which appears to be an Australian film or TV movie from the 80s with a sci-fi Rear Window like plot, but I can't find out much about it.
I think this could be Headless, a Channel 5 film from 2000.
It's way before 2000. Some time in the 80s is all I can remember. I'd have caught it on TV when I was a kid / early teens I think. Mid 80s maybe.