This is something of a nonsense problem that some had made a big issue last year. Just call it the 8th British division - I don't think anyone, anywhere, except maybe the most anally retentive would demand 8th UK division!
I agree with 8th British division, but I'd suggest that a common usage might well be 1st Division (U.K.) alongside 1st Division (U.S.).
On further research, 8th British Armoured Division seems to be the British usage. Long usage may make that an honorable exception. Military writers would tend, in disambiguating, to say 1st Armored Division (U.S.), 1st Tank Division (U.S.S.R.), 1st Armoured Division (U.K.). While 4th Canadian Infantry Division was standard enough, U.K. vs British and American vs. U.S. would occasionally clash as naming conventions.