Welcome to the Forum.
Always lots of help and advice on here. Ignore grumpy old men, but not me!
Since the Milling machine runs, you have power coming into the shop.
The fact that the lathe runs if you hold the contactor says that you have power to the lathe, but suggests that something has gone open circuit in the contactor circuit., possibly one of the coils, since it is on 3 phase.
You need someone who is familiar and skilled on 3 phase circuitry. Unless you know what you are doing, I would strongly advise against trying to trouble shoot a live 3 phase circuit.. You have upwards of 400 volts around and a fair few Amps available, which could be lethal.
If the machine were disconnected from the power supply, you could check each of the contactor coils for continuity.
other than that, leave it for someone who knows what they are about to find and correct the problem.
Good Luck!
Howard