Electronic – arduino – Replace mechanical pot with digital

You ought to consider that the volume pot is logarithmic and this might mean that the digipot's resolution needs to be quite high to get a non granular change in volume when you control it.

Probably not, practically.

It sounds like you have thyristor phase controllers and the pots would not only have to be inconveniently high voltage and power, but also you'd need to add isolation from the mains for safety.

You'd be better off (and much safer) finding a motor speed control that has an isolated low-level control input.

Edit: Speed controls of this type energize the field coil (if any) with a constant voltage and control timing of pulses to the thyristors supplying the armature based on the input speed command signal and motor current (to maintain a more constant speed). By controlling the timing relative to mains zero crossing, the average armature current can be controlled, and by measuring the armature current (a proxy for torque) the speed can be regulated fairly well without a tachometer.

For a controller with an external low-level input it would be designed from the ground up to use low level signals and something like an optocoupler or a small transformer would be used to transfer the control signal (perhaps 0-10V) to the mains circuitry using a linear optocoupler or a PWM scheme.

Trying to add a digital pot to one of these things (in my experience) would be like trying to add a digital pot to a wall light dimmer- really totally unsuitable and you're better off starting from scratch, with a much more complex design.