James,
To put it in context, I use the 220W motor to drive a Peatol and it is more than enough (1/3 HP on a 2.5 in lathe!).
From what I can tell, your lathe is a clone of the ubiquitous Chinese Mini Lathe (I have one of these too)
If it were for me, a 500W motor should be more than adequate for the CML (looking at the motor picture, it is actually rated at 3.6A which if true means the maxpower may be 30% higher 
Without having one on the bench, I think it is as good as anything currently supplied with this class of lathe.
Check the motor size to be sure it will fit in the tight space
Check the no load RPM x 0.85 or so against the ratios on the drive train to be sure it gives you the desired range of speeds.
The controller supplied is OK but simple, you can get better ones – did see some being sold as plain boards with options for ramp and torque and electronic reverse for about £20, but cant find these anymore, or you can fork out £100+ for an American made KB controller (but from what I have read, these aren't much more reliable than bog basic controllers from china)
If suitable either get the 500W motor and PWM drive package or a more powerful motor if it will fit and you really think you need it, but beware of over motoring the lathe. To make life simple make sure to get a permanent magnet type brushed motor with a suitable speed range. Find whatever controller suits.
Rip out all of the old electronics from the lathe control box.
Check the remaining switch gear – does the mains on off/e stop switch function and more importantly, does it incorporate a latching servo (ie is of no volt release type)
Secondly, does the rotary switch rev-off-on look butch enough to handle about 10 A and is it at least 3 pole? – you will need to check carefully it can be used in a dual pole dual throw manner by working out the connections with a multimeter (or a battery, bulb and some wire as a simple continuity test)
Because on my CML, all of the switching functions are latched electronically, I suspect the answer is no to both of the above. You then have a choice – If the switchgear is still functional from checking with a multimeter, visual inspection etc use them anyway and devil take the hindmost until they fail or the fuses blow (but remember to always switch everything off after use and check that everything is always off before plugging in) Remember also that you will be wiring the DC motor feed of 240V via the rotary switch so make sure you are content with mounting, isolation, robustness etc)
Alternatively, if in doubt, or unable to make a judgement, go to the usual outlets and get….
**LINK**
and
**LINK**
or search for 10 amp DPDT toggle switch
and because you never know when you might need it, a cheap BIG RED BUTTON type E stop switch, all together new switchgear should cost less than £20
and wire as follows – Estop – NVR on off – controller – DC out to Rotary motor run reverse switch – DC motor
remember this will be a manually switched system, so you will have to remember to avoid casually flipping from forward to reverse without letting the motor stop and also avoid starting with PWM drive set at maximum speed – just to be kind to the motor and drive!
If all that is too daunting, then try UK lathe suppliers that carry spares such as Amadeal, ARC eurotrade and purchase a complete motor and control board set up but you will be looking at well over £200 outlay