Or shot just after he got out of bed in the bedroom?
No, because the blood stains all belong to June. Plus the bullet entrance wounds are at an angle meaning the shooter was at an elevated position.
Plus If Jeremy had Neville at such a point blank range he would have just put a shot between his eyes or forehead and that would be the end of it.
Knowing this I have devised my own theory on how the events would unfold had Jeremy been the Killer.
1- Jeremy enters via toilet window, Then he walks towards the gun cupboard to get the gun.
2- While Jeremy is getting the gun and ammo from the cupboard Neville wakes up from Jeremy doing this so he gets out of bed and starts to walk quietly downstairs via the main staircase.
3. Unaware that Neville is awake and walking downstairs, Jeremy then goes quietly upstairs via the staircase over the gun cupboard once he is all loaded and ready
4. Neville then walks around downstairs trying to work out what waked him while Jeremy is walking towards the main bedroom upstairs
5. Then Jeremy shoots June several times and was also expecting Neville in there also. Hearing the shots and screams Neville rushes upstairs calling June
6. Jeremy realises Neville is coming upstairs so he is goes towards the bedroom door where he sees him almost up the stairs and the shoots at Neville forcing him to retreat back downstairs towards the kitchen.