If, and only if, Sheila is the perpetrator, I agree with a lot of your post Sherlock. I think that no shooting had occurred when Nevill rang Nevill, and that Sheila was downstairs. I don't know why Nevill didn't get the gun off her, but perhaps she was pointing the gun at herself so he didn't want to risk it.
I find it easy to picture Sheila keeping Nevill away from her by pointing a gun at him ...
But it is a good point - she may have been threatening herself with the gun ....
Maybe Nevill did not think it would escalate ....
He thought it was obviously time for Sheila to be sectioned back to hospital ...
He thought that by phoning Jeremy that Jeremy would phone the hospital to come and collect her ...
He probably did not want Sheila realising that the hospital would be on it's way to take her back ...
But even the call to Jeremy was enough to trigger Sheila to escalate things ...