Don't think it can be a Commonwealth War Graves Commission stone as they only commemorate the dead of the two world wars. Someone must have commissioned a stone that looks like one.
Freshly dug could suggest another burial has gone in recently.
There are four official war graves in St Peters Churchyard:
http://www.cwgc.org/find-war-dead.as...name&order=asc
[Edit to add: That War Graves Photographic Project collects post war MOD stones as well which have a different shaped top -That is possibly one of those.]