Another cheapo solution:
If there's somewhere on the spindle or its pulleys where you can attach a little magnet , buy a "bicycle computer" or "cycle speedometer" for 7 or 8 quid (I'm avoiding the pound sign problem) and set it to read in kph rather than mph. You can also program in the circumference (not diameter) of your fictional bike wheel. If you go for 1667mm, the display reading will indicate 1/10 x the rpm. The reading shows to one decimal place, so 99.9 kph equates to 999rpm. Many won't go above 99.9kph, but Halfords sell a model (ASIIG, I think) which will go up to the improbable pushbike speed of 200kph.
I only mention this because the sensor (intended to clamp on the front fork of a bike) and the read head can be left permanently attached to a lathe. The actuating magnet is meant to go on a wheel spoke, but if you can't find a place for it on the lathe spindle or spindle pulley, embed it in a cork which can be stuffed into the outboard end of the spindle when required.
Andy