You could use a modern electronic (thyristor) foot controller for sewing machines (instead of the old graphite pile types), which at least is unlikely to boil your foot. Or purchase a cheap (£5 to £25) generic ac motor controller from the usual on line markets and wire in
There is no plug and play brushless unit available that I am aware of unless you want to get one of the far east 500w to 1000w servo motor and control packages (circa £100+ and probably too much for a watch lathe). Some are now adapted for lathe use rather than sewing machines. As usual, no backup.
If you scan about on the net, sometimes you can locate small (but not toy)30 to 60v 100 to 400w brushless motors that run at a reasonable speed, but you are on your own as far as sourcing and rigging up power supply, drivers and controllers. Some claim to not require a hall effect controller, but as usual with Chinacarp, they provide no useful information.
Brushless CNC spindle packages are available, but are likely too high RPM for lathe use.
There are some nice 1/6 Hp 3P induction motors on ebay for £30 delivered at the moment, but you will have to get a a VFD for £40 and wire it up yourself.