Solved!!
Thanks to a chat with Dan Booth from Nostalgic Motors. My problem was under the hole in the striker, where it faces the lock mechanism as the door shuts.
There was a small groove worn into the chrome plated brass of the striker, where a corresponding tab on the mechanism comes into contact. I've ordered two new strikers from Dan - but in the meantime, I drilled a tiny hole thru the striker in the exact spot of the groove and screwed a tiny screw into it. My doors lock!