Terry, hopefully at least a later three-rail gearbox was adapted for that 1500, or the owner must be a bit unhappy. The 10/early Herald gearbox had almost a "granny gear" for 1st; in fact, some of the owner's manuals actually suggested that 1st was not necessary for normal level starts from rest. (My dad always drove the 10 that way and then quickly ruined the first clutch in the new Herald 1200, which had different ratios and was NOT designed to start in 2nd normally!) Also, on those early boxes, the gap from 2nd to 3rd was enough to drive the proverbial truck through...again, fine for the early motor but not for the later, torquier ones.
The Herald/Spitfire gearboxes mount differently in the rear, but it shouldn't be too difficult to adapt something there. An all-synchro three-rail Spitfire box would be ideal, I'd think.
As for the rear axle, it was a Standard unit, first developed in the early 1950s for the Standard 8 (803cc) and continued through the 10s and Pennants and commercial versions of same. My understanding is that the same axle continued to be used in India, where CKD Heralds were built under license by STAMPRO. That company later redesigned the Herald into the Standard Gazel, which reverted to that live axle rather than the swing-axle setup of the Heralds.
Seeing as Riley was part of BMC, somewhat of a rival to Standard-Triumph at the time, it's unlikely BMC would've willingly shared such a major component with the competition except at great cost! (Heck, that's why we have Zenith Stromberg CD carburetors: Triumph designed these in the early 1960s to avoid paying BMC the high prices charged for SUs! Ironically, Triumph continued to use SUs on the Spitfires, even though there were 1.25" versions of the Stromberg! On the other hand, the last MGs in the US all used single Strombergs.)