I wasn't able to find a military register online
There isn’t any register or similar online for WW2.
But FindMyPast has
World War 2 Allies Collection
British Army Casualty Lists 1939-1945
Royal Artillery Attestations 1883-1942
Army Roll Of Honour 1939-1945
And various other lists. You might be able to use them to at least make your list of possible men shorter. because presumably only some regiments were in the Netherlands.
I would look for names in the newspapers (also on FindMyPast) and The Gazette.
Also, even though you have narrowed it down to Shropshire families, the descendant could well be in a Commonwealth regiment or even the Navy.. my own father-in-law, (a New Zealander of Scots descent so we can rule him out) was commander of a landing craft on D Day. Among their various tasks after D Day they went up the coast and delivered supplies along canals in The Netherlands. He and his crew were Volunteer Naval Reserve from a mix of countries.