It's 5/8 20 for sure. BSC I'd say for the lack of other matches. Problem with turning down is that the clutch push-rod on the Burman is quite a chunky thing and there isn't that much metal to play with if you clean right down to the minor diameter to start again. No chance of getting it back with a die and a new nut?
Faced with the same problem a year or so back, I cleaned things up as best I could, cut the thread just a tad deeper in the lathe and made a new nut a fraction under-size. Hell of a palaver but works fine.
Good luck!