This was originally fitted to the twin carb version Coopers and "S", with CHOKE AND LOCK written on it in recessed white paint.
MK1 & MK2 Minis share the same design choke cable, it has a plain black plastic knob with the word "CHOKE and LOCK" on it.
Common sense says you can use this on any Mini.
If yours no longer locks on or the cable end is hopelessly frayed, then replace it when you can.
The grommet in the body for this cable to pass through is CAM5969 if the hole is 11/32".