1/16" isn`t a huge amount of movement(although Dad has just checked and there isn`t a measureable amount on ours), does it correspond with slack between rack and pinion gear? You may be able to take up the play in the pinion on the endcap, below:
...And play in the tie rod inner joints can be "shimmed out" by separating the joint and removing shim or shims to suit:
Both jobs can be done without pulling the rack, although getting the tie rods apart is a pain, if you don`t have the correct tools; I ended up having to hold the base of the tie rod in a lathe whilst turning the cap with first, a C-spanner, and then Stilsons!
The Morris Minor rack is fitted to the Minor with the pinion on top of the rack, but on the Greyhound, the pinion is underneath, so is it a LHD rack fitted upside down?
Hopefully see you at Goodwood!