The easiest way then might be to get the external E-stop kit, and re-wire it to use a relay to your motor connectors.
It looks like active-closed optocouplers, so you can wire all the motor outputs in parallel (+ to + and - to -) and any one that goes "on" would make the estop go "on."
https://tormach.com/e-stop-interface-kit-30785.html
Even if you don't buy their hardware, you will get some useful information from their installation instructions.
Get a DPDT relay with a 5V or 12V or 24V coil (24V may be better, as it will draw less current) and contacts rated for 240V/20A. Pull +24V (or what you choose) from the power supply in the mill, to the "+" of your ganged alarm outputs, to the coil of the relay, to ground. Wire the Estop addition as appropriate (see the instructions) to your relay.
Note that this would be an "active stop" relay -- when the relay power fails, it defaults to "no stop," which is probably good enough for what you want to do. If you want a "fail open" mode, you would have the invert the outputs so they are "normally closed" and flip the relay wiring appropriately.
Also, there's no current rating on the output optocoupler specification. If the relay coils want more current than the couplers are rated for, you'd need to either use a solid state relay instead (always a good option for reliability) or a driver circuit for the relay so the relay can drive the estop. A right pain, that. But, in theory, the electronics are simple and straightforward.