Hi
WW2 records are still with the MOD at the moment but are in the process of being transferred over to the National Archive.
If he was in the artillery his service number may have been 959051 as there is a recod on FindMyPast for a Harold Arthur Forrest who joined up in 1939. This is in the follwoing collection of documents
"Royal Artillery Attestations 1883-1942"
Unfortunately it doesn't say much more but if you have his birth date and service number your Mum can try to get his service record.
Nesta