I know virtually nothing about tracing WW2 soldiers but if it's anything like WW1, then a good start for the guy in the 5th Battalion, Black Watch, would be to find out what that battalion was doing on the day in question. Which, in the case of a WW1 battalion would mean reading their War Diary. His father, in the REME, might be more of a problem since you don't have a specific unit.
Either way, the other thing to do would be to apply for their service histories from the MoD - a lengthy process. I am not clear either on confidentiality rules (I'm sure they keep changing) but generally it's easier (and you get the full story) if you are a descendant (and I mean "direct descendant", not a "relative").
Adrian
PS - I'm quite expecting someone to say "But you can't access WW2 War Diaries easily!"