I don't have a brake booster, but I imagine it should fit. The wiper motor does not stick out past the firewall cross-member (as seen from this angle) and a right hand drive car would have had master cylinders here where my heater hoses come up through. In this photo you can see the brake lines going to the brake valve and the clutch line that passes through a bracket, both right in front of the cross-member.
Post a picture or two of your engine bay, showing the position of your brake booster. Maybe it was not positioned properly when your car was converted to left hand drive.

Thanks Don. I have a brake booster on the RH side of the firewall. Must have been part of the LHD conversion. I'm afraid I won't be able to get a wiper motor to the RH side of the car.

Photo - wiper motor in later left hand drive Mini. Ignore the fuel pressure regulation system. There should be lots of space in your MK III car. Wipers park to the right, so the pivots go in the left pair of holes. As built, Minis from this era had 4 wiper pivot holes, with the unused ones plugged with rubber. In my photo, you can just see the black plug to the (photo) left of squirter and the base of the right wiper to the right (car left).

You should not have to alter the wiper park mechanism as it should park with the wipers pointing toward the wiper motor. You will also need left hand drive wiper arms, or bend the tips so the blades park level with the bottom of the wind screen.

I need your email  to send you photos. btw. I have couple new wiper motors for some reason I found recently when I was doing my mini pick up resto..

Thanks Donnie. How about sending me photos of a proper RHD with the wipers in parked position and the location of the wiper motor so I know what not to do.

 you Should have left it as  proper RHD.  then you would Not be having any problems.

I am looking for photos of a LHD MkIII that show the wipers in parked position and also the position of the wiper motor. My car was converted to Left Hand Drive in the past and I think the wipers were never done properly. There is not a place for the wiper motor to be located on the right hand side.