One obvious solution to how to make it work would be a simple micro controller, which would sense which selection button had been touched, read the proximity sensor, and run the appropriate motor for the required length of time to dispense the "look beautiful juice".
Problems:
What is the power source?
Mains supply in a bathroom is awkward with UK regulations.
Batteries, will have a short life driving motors.
Environment.
A bathroom will be humid, lots of scope for water condensation on the electronics.
Have a look at electric toothbrushes to see how they do it.
JimB