May be stating what you already know, but -
HLW 159 was new to London Transport in September 1947, one of the earliest AEC RT buses delivered after the war (the prototype took to the streets in 1939, and a small batch was built during 1940/1 before production was stopped because of the war.) It carried fleet number RT 172 and was new at Croydon garage.
By 1958, London Transport had too many buses - bus use had been growing until the early 50s then declined quite rapidly for a variety of reasons including increased car ownership, population in inner London not returning to pre-war levels, a spiral of staff shortages, unreliable services, and a long strike in 1958.
In 1958, LT started selling off the post-war RT fleet, and Bradford Corporation was one of many operators who bought some, HLW 159 became number 410 in the Bradford fleet, running until 1968. It entered preservation, and is one of few RT buses that went 'second hand' to another operator to have been preserved in their livery rather than returned to London livery.
Vehicle history from
Ian's Bus Stop website.
The vehicle that went to Bradford isn't the same 'kit of parts' that entered service as HLW 159 - London Transport's overhaul process included separating chassis from body, and the same set was rarely returned to the road together - they also had special dispensation to change vehicle identities, to avoid de-licensing and re-licensing buses, so a 'bus' (as in registration number) would come in to the works for overhaul one day, and a newly overhauled bus would go out the other end of the works with the same identity within a day or two.
More about Aldenham Works and the overhaul process on the '
Red RF website'