My mini lathe. The white plastic things in the foreground are the high speed / low speed gears. I stripped a couple of teeth off one. To fix it you need to completely dismantle the lathe to remove the headstock, press out the spindle, press out the layshaft and reverse. I don’t have a press so I bought 2 pieces of steel plate and used 12mm studding to make a press. A bearing puller and new Thor aluminium / nylon faced 2lb hammer completed the job.
Along the way I found out just how badly the lathe had been assembled.
The white plastic bits have been replaced with steel ones, all the bearings replaced with brand name ones (including angular contact bearings for the spindle, top tip, check which way round they go before fitting them to avoid having to do the job twice). If something breaks now it’s likely to be the motor pulley or belt, they’re much easier to replace.