You've really thought this through... I must say, though it's awful to even think about let alone read, I tend to agree with this. NB was more than likely the first to have been murdered.
In all likelihood June was dead before Nevill. The initial shot to her head was quiet severe. She walked around the bed then collapsed before getting out of the room. The shot between the eyes was probably delivered after she was already dead, it was simply to make sure. If she was still alive and Nevill died before her then the kitchen fight and reloadings were rather quick.
The gun reloadings were worked out a while ago based on the totality of the evidence including spent casings.
Julie said that prior to the murders Jeremy planned to say he got a call from Nevill. Such plan eliminates the ability to kill Nevill in his sleep. How could he claim Nevill called if he died in his sleep? June was attacked in her sleep though. At least 4 and possibly 6 shots were delivered as she was lying in bed. The 2 to her chest could have been as she was sitting up in bed or even as she started to get up. So he most likely fired at June first.
Nevill was shot as his left side profile was to to the killer. He could have been sitting in bed or could have been getting out of bed. He definitely wasn't upright because the wound to his jaw and arm were downward trajectories and if he had been upright then the trajectories would have been up. Vanezis failed to consider that Sheila and Jeremy were both shorter than Nevill and thus would have needed to aim up if Nevill had been fully upright.
The magazine only holds 10 rounds so this means the gun was likely empty. There is a way to load 11 by chambering a round and then ejecting the magazine adding a 10th round and then reinserting it into the gun. If there were 11 that means the shot between June's eyes was delivered as well at this time. It seems odd though that she would run around the with the killer still there in the way. So I am more apt to think the shot between her eyes came later. Once the gun was empty one of two things happened- 1) the killer fled to the kitchen to reload and Nevill chased the killer finally catching up in the kitchen or 2) Nevill fled to the kitchen to arm himself with the killer chased him and caught up in the kitchen.
The whole reason why things proceeded to the kitchen is because Nevill wasn't immobilized by the 4 shots he suffered and the gun was empty. Nevill was beaten until he was unconscious and the killer was then able to reload. He was then shot 2 times in the top of the head and the killer moved slightly and then shot him 2 times in the right side of the head. The killer then fully reloaded and went upstairs again with either 10 or 11 rounds in the gun. If 11 then at this point the killer shot June between the eyes to make sure she was really dead then used 8 shots on the boys and the last 2 on Sheila. The gun was then empty again.
So the gun was partially reloaded to kill Nevill in the kitchen then fully reloaded with 10 rounds combined used on Sheila and the boys. in all likelihood the boys were killed first then Sheila bu if Sheila had woken up and been out of her room then she was killed before the boys. There is no way to know for sure if she woke up herself or not. I doubt she did given her state of being overly tired because of she was being given too much Haldol and wasn't taking a countering agent.