They used what was fitted and what was available. Period photos show predominantly either the Dunlop Universals or the Good-Year 'All Weather'. Quite often bald as well.
I do have a pair of modern Continental Twinduro TKC 80 I thought these might be suitable for typically good grip in sand over desert terrain as during the war.
This is from the wartime "Data book of RAF vehicles", It covers all "B" Vehicles including motorcycles, the sand section look pretty similar to some of the original motorcycle tyre patterns which survive.
The first SWD Norton Big 4 I bought in 1973 was painted green over desert sand and that came with Goodyear Sand Tyres - all bald I might add but for £10 although not complete it was hard to complain.