I have Neville being chased downstairs, rather than him pursuing the killer.
That's what I used to think or assume Hartley. I'm not sure where the notion came from in the first place, but wasn't there an assumption that Nevill had been shot on the stairs which would imply that he was shot from behind whilst going down the stairs?
Now that there's confusion about the empty cartridge on the landing, I'm not at all sure that he was shot on the stairs.
I'm now considering that it was the killer who went down the stairs because he/she ran out of bullets, and that Nevill followed.
How quickly could Nevill have gone down the stairs? If the killer was Jeremy, surely he would have caught up with Nevill before he even got to the kitchen.