Have 3 machines on 1 wire
In general upgrading the wire is an acceptable solution, but you will probably also need to have new breakers at the laundry end of the wire.
You will need to put a new breaker box in the laundry at the end of the wire and have breakers for your three appliances there. there are probably rules about which parts of the room are allowed to have the breaker box.
Alternatively if your house has a three phase service you could just run two more lives in the conduit (from the other two phases), and share the existing neutral and ground wires, that's assuming there's room in the conduit for two more wires.