I don't see why not a forum meber was looking into it a while back i'm sure a TR engine has been done into a Mini and there less space there. I know a few Rover V8 mini's have been done so a Mk1 with a smaller engine should be fine.
All you'd need to do is make the mounts, driveshafts, radiator piping etc and a stand alone/the old ECU (stand alone ones are probably easier with newer engines) Then the Mk1 electreics are fine for indicators etc.