The starter is a Crabtree B15 with an oil filled dashpot adjustable overload at the top left corner.
Both the fixed and moving contacts are replaceable, I have some spares in the depths of a cupboard.
The external control buttons as follows : stop button operates on the top right red shaft and the start operates the bottom contact block which has the contactor hold circuit connected.
Standard fitting on most Colchester lathes during many years.
Looing at your top picture it looks like the suds pump supply is linked to the motor feed but there may be a separate suds pump starter for control as you don’t always want coolant flow.
Emgee