Servos do sound like an easier approach to get working, but I'd have thought that any existing valves/controls would be inside the shower cubicle and that's a pretty hostile place for any electrics/electronics.
Perhaps you could approach this by using the Arduino to control valves in series with the existing mixer valve / taps rather than control the valve/taps directly? I assume that you will be concerned about the possibility of scalding in which case a sensible approach IMO would be to design your system to restrict the hot water flow to keep the output temperature below the target. This also has nice safer failure modes - you always have an unrestricted cold water supply and always have the option of shutting off the hot water using the existing manual controls if something goes wrong.