There was blood all over June's pillow.
The evidence is she was shot 5 times in bed.
I am not being dogmatic about it. I do disagree with you, but I also acknowledge that you may be right. Something in your favour is that it was a low calibre weapon.
However, for you to be correct about this, that would mean June is getting up and out of bed when she is injured and while the assailant continues firing at her. Remember where she is found and in what position: on her back by the bed with her feet near to the door, and with blood around the bed, all of which suggests she has been moving around. Consider her age. She is an elderly woman, and she may be in shock.
Bear in mind also the psychology of this. If Jeremy is the killer and he has caught Nevill and June still in bed, then he will shoot Nevill first, not June. I think you undermine your own case when you try to suggest that Jeremy is firing on June first. If Nevill is in bed, then it's easy for Jeremy. He has plenty of time and can walk up to Nevill and shoot him at close range, then shoot him some more as part of the staging to make it look like Sheila run amok.
Finally, also take into account the position of the brass bed itself. it is at right-angles to Jeremy's likely route of assault.
Taking all this into consideration, I find it more likely that a Jeremy scenario involves Jeremy negating Nevill downstairs (Nevill having decided to sleep downstairs, for the reasons given) and Jeremy then goes back up the stairs and finds June on the landing. As June backs into the master bedroom, he fires on her. She backs into her bed, where she leaves more blood, then comes about two feet forward towards Jeremy, probably screaming. Then he kills her by shooting her between the eyes (this last shot occurring either there and then as just described, or later if he finds her still alive).
I think the police then moved June's body. Her body was probably originally found with her feet somewhat at the threshold, Jeremy firing from the stairs, then the landing. The reason I think this is because if you look at the poor woman's body in the photos, the position she is photographed in doesn't make sense. It looks like her body is laid flat and has been 'staged' by somebody.
This all fits the evidence - if you assume Jeremy is guilty.