I was under the impression it was the same transmission with the side gear change...and that on the AH 1st was locked out. My info could be incorrect, however, here is a bit of it:
Production Austin-Healey 100s were finished at BMC's Longbridge plant alongside the A90 and based on fully trimmed and painted body/chassis units produced by Jensen in West Bromwich. The first 100s ("BN1"), were equipped with the same 90 hp (67 kW) engines and manual transmission as the stock A90 but the transmission was modified to be a three speed unit with overdrive on second and top.
Healey had been using the big BMC saloon running gear, in the 2.6 litre four cylinder Austin Healey 100, and in 1956 adopted the 'C' series to make the sports car the 100/6. This was the Austin A105 engine with the Wolseley twin SUs and 102bhp, giving excellent performance. The cylinderhead was soon improved with a decent manifold, and power grew to 117bhp @ 4750rpm.
In 1954 the new Wolseley 6/90 was introduced with a new BMC twin SU Six cylinder engine, called the 'C' series, of 2693cc and four main bearings. It used the same 89mm stroke of the 'B' series, and also shared its timing sprockets, chain, and cover, with the smaller 'A' and 'B' series. This 95bhp Wolseley was a fast car, and many Police Forces purchased them. The sister car, the Riley 2.6, gained the same engine in 1956. Prior to the better known Wolseley, a Morris 'Isis' six cylinder car was shown at the 1954 Motor Show. It had an 85bhp 'C' series engine with a single SU carburettor.
The new engine was a product of Morris Engines, not Austin, and things like the camshaft being on the opposite side to the 'A' and 'B' series, a proper Reynolds chain tensioner, as well as solid little ends, show the design differences. However, Morris Engines had put the cylinder head holding down studs through the cast-in-situ inlet manifold, not good practice for good airflow. This six cylinder engine gave the big BMC saloons the edge on their Ford and Vauxhall competition, as it had more torque, even though the Austin A90 version in late 1954 still used a down-draught Zenith carburettor.
It was about this time BMC were improving the Riley Pathfinder that used the high-camshaft Riley four cylinder engine. The replacement model was to be called the Riley 2.6, and used the BMC 'C' series. An M.G. version was almost made as a development car, but never taken seriously. More interesting was a 'Z' Magnette fitted with a 'C' series engine giving very good straight line performance.