Depends on which spindle control switch is fitted to your lathe. There is a live mains voltage AC wire to the common side of a NO (normally open) pair of contacts, this linked to the other Common terminal so there are 2 sets of contacts working in parallel.
Then when you operate M03,and the relay energises, a cable from the linked NO contacts is the live AC feed to the Motor Control Board to start the spindle.
Basically the relay is in parallel with the switch so either will start the spindle regardless of position of the other device.
If you have the latest 3 position NVR switch fitted check the schematic drawing for further information, probably best to do that in any case as the CIRCUIT WE ARE DISCUSSING IS AT MAINS POTENTIAL SO BE SAFE AND REMOVE PLUG FROM POWER SOCKET BEFORE WORKING IN THE PANEL.