Dis he belong to a Regimental Association ? If so, they may be able to help.
He would have been awarded campaign medals and these would usually have his service number engraved on the edges. Speak to his close men friends, especially those he knew for a long time - the may have shared memories which might give you important leads, ie, regiment, company etc.
The fact that he went on to serve in Burma after fighting in North Africa and Europe will narrow your search considerably as not many units did this, unless he transferred regiments. Once you have discovered his regiment there will be websites you can access for more leads.
The essentials are regiment and service number.
Good luck - and commiserations.
This is not a criticism, but children often regret not talking to their parents about their past. Once they've gone, their history has gone with them.
D. Ex 3 Para.