Triumph experimented with a Borg-Warner 35 back when the TR3 was still current. Later, the six-cylinder Triumph 2000 sedan had essentially the same gearbox, and at least a couple of folks have converted TR6s and six-cylinder Vitesses with those. Keep in mind that the 2000 sedan, in manual gearbox form, used a slightly modified TR gearbox...and you can see where I'm going with this, I hope.
There likely are much better and more modern autoboxes that could be adapted. For example, there's the mid-1980s Volvo four-speed automatic, which I think was a Borg-Warner 65. I don't know much about the physical size of that one, but the older three-speed 35 was a much wider box than the original TR manual gearbox; that's just one thing to keep in mind while ignoring all the ancillaries needed to make an autobox work. Realistically, an engine/autobox unit swap might be easier to pull off?