to see the gubbins I would have to take the machine apart which I'm not sure I want to do until I know I can commit to this project (i.e if im gonna break something, I want to be sure I have a project in place to fix it / ammend it!)
Is your plan to move the speed switch with a motor, so you don't have to take it apart? That probably could be done with a servo motor or a stepper motor. Otherwise, you can't get started, or know if it's even feasable without taking it apart an figuring-out how it works... Even after you disassemble it, you may not be able to figure-out how the speed control works...
If the speed control operates smoothly (like a volume control) a small servo motor could work. (It reqires less electronics than a stepper motor). You'll just have to figure-out how to attach the motor-shaft to the speed-control shaft. If it's a multi-postion switch, a stepper motor is probably best. You'll need to measure (or estimate) the torque required to turn the switch, and then select an appropriate stepper motor and stepper-motor driver-board.
Operates on A/C power, 250 volts mains powered, (EU converter plug required)
Without knowing otherwise, we have to assume that there is 250V going through the switch. You can't connect that much voltage directly to the Arduino, and you need isolation for safety. Relays are the eaisest way to get isolation and a relay is a (electrically operated) switch, so it's the perfect replacement for a manually-operated switch. Although there are rotary relays, finding one that matches your rotary switch would be impossible or too expensive, so you'd have to use multiple "normal" relays.