Which 5 speed are you looking for? The 56, 60 or 63 series?

I have a 63 series that I'm not using and a 60 series in the car. My intent was to go to the 63 series this summer so the 60 series would be available. With the 63 series box you'll need to find an automatic driveshaft as the yoke for the 63 series is larger than the standard 1200 yoke.

As for the Austins built under license those motors have absolutely nothing in coming with the A-series engine. It's akin to comparing a Ford flat head V8 with a 302, they are that far apart.

There is / was a custom bellhousing available for fitting a Toyota T50 box but here in the USA you can find a 60 series box for much cheaper.