There seems to be quite a bit online about this situation, though not exactly what you're after. For example, there's an article on http://www.centenarynews.com regarding the admittedly unusual execution of an officer for cowardice which says, 'Eric Poole was executed at Poperinge, near Ypres, in Belgium. He was 31. Original records state only that he 'died of wounds'.
This individual has a grave and headstone, which further indicates the confusion there was over this type of 'crime'.
There should be records of Courts Martial stored somewhere and these would indicate actual sentences passed. I doubt a Death Certificate would be entirely truthful as it would have been available to the public by application. Listing 'execution' as cause of death wouldn't have been good for the country's morale at that time.
You might have better luck joining one of the WW1 forums and searching and/or asking there.
Don't know, but if you send off for his death certificate it might say how he died. If it is not on it then it would probably mean the family were not informed.
I think that maybe if it was a court marshal the family might know as it would probably be made public. If he was shot for cowardice, this may have happened 'in the field' situation in which his death was just as a result of injuries.
All maybe's and don't really knows, sorry about that.
Ps anyone can send off for someone's death in the UK. It was the only way I could find out how my boyfriend died, as his relations were not forthcoming.