Hello.
I think that the rotation direction might be controlled by hand, using a lever or a switch since mechanical lathes are turned on and off manually (including the rotation direction) could be a good idea.
The speed control could be carried on using a AC dimmer. If the AC motor is a single fase one, thing I think is true because washing machines are designed for a single phase voltage line, you can control its speed with a really simple circuit.
Just search "ac motor dimmer" on any search engine and you'll find a lot of circuits; take into account that as an inductive load is being used a properly capacitive compensation should be calculated and used. There exist several C-L phi cosinus compensation calculators, maybe the motor has connected (and should have) its compensation capacitor, if not, look for it at the PCB and unsolder it or buy an equal one.
Just ask if you didn't understand anything.