We're kind of getting into the sort of territory that joining a club would be so useful for now which is "what else do you need" because while we're on the subject you've also got to factor in safety (lungs and eyes/face) and sharp tools (many options but at a minimum a bench grinder is essential as the tools are HSS and blunt quickly).
Sharpening and turning are marriage partners in the same sentence. Turning without sharp tools is simply not possible and furthermore is dangerous.
So with that said here's my starter for 10 shopping list:
- Lathe
- chuck (can do without but wont want to)
- tools (roughing gouge, spindle gouge, bowl gouge, parting tool, skew chisel at a minimum)
- sharpening means (dry grinder at a minimum)
- health and safety (face mask or powered respirator, shatterproof eye wear, dust extraction or at a minimum a good quality dust mask)
- calipers for measuring diameters (could be a knackered pair of Aldi verniers with the corners ground off)
- decent drive and live tail centres (pref steb type but again not necessary at the start)
Personally I would also strongly coach you in the direction of variable speed. It's so very very very very very very very much better than manually changing a pulley belt.
So just make that £200......£800 and you'll be fine