Unless he was injured, listed as missing or prisoner of war, it's highly unlikely there will be any official records for him on line. The service records of men who served in the Army after 1920 were still with the MOD but are in the
process of being transferred to the National Archives. Depending on where his records are, there are two different processes you need to follow to get a copy of his record. I suggest
starting with the MOD, and if they categorically tell you his records have gone to Kew, then approach TNA. The MOD is only interested in proof that the subject is dead, and it can take a long time for them to process requests;TNA on the other hand treat requests as Freedom of Information requests and have arbitrarily imposed a time limit which means they don't release records until 115 years after the date of birth of the individual. That said, if you are able to visit in person, you can look at his record and take photographs of it etc. As you mention a search of the currently indexed records at TNA does not bring up his record, but that doesn't mean it isn't there, just that they haven't gotten around to indexing it yet.
If you know anything more about which Royal Armoured Corps Regiment he served in you can also access the war diaries for that unit, and follow his unit through North Africa and possibly on into Italy in 1943. However he won't be named in the war diaries.