Wow, this is a "strange" request.
Oddly enough there is a possibility of converting to an auto transmission. Several other cars used the BMC C series engine & some had the Borg Warner type 35 three speed auto transmission. A transmission from an Austin A99 or A110 would likely adapt as would one from the Wolsley 6/90 & 6/110. These models are still around in Canada & Europe but few in the USA. Also a few of the the Triumph Stags used this transmission but I doubt if it would adapt as easily as the Austin or Wolsley unit. Someone more knowledgeable than I will have to advise of the details. I doubt if many have put an auto behind the standard Healey engine, but you never know.
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Here is a partial quote from another web site regarding the BW Type 35 auto.
"On all Mark II Super Minxes a new Borg Warner type 35 automatic replaced the Smith’s Easidrive of the Mark I. A little history of the Borg Warner type 35 transmission is in order here. Borg Warner was a major supplier of gears and manual transmissions to the American automobile companies. However, their biggest customers were the “independents”- Studebaker and American Motors; GM, Ford and Chrysler generally used their own transmissions. In the early 1950’s Studebaker approached Borg Warner and asked them to develop an automatic transmission that was more efficient then the units available from the “big three”. The reason for this was that Studebaker’s strongest selling point was good fuel economy, and this was achieved by using small engines- a 169 cubic inch six producing 85 horsepower. This engine did not have any power to spare for a transmission. American Motors Rambler engine was only slightly larger and also produced only 85 horsepower. Both Chevy and Ford’s smallest engines developed well over 100 horsepower (Ford’s six reached 145 horsepower by 1957). Borg Warner developed the Type 35 (the three indicates 3 speed-there was a later Type 45 transmission that had 4 speeds) for Studebaker and American motors."
All of the auto conversions that I have seen in the USA were behind a V8 engine transplant.
D