Bizarrely the lathe motor has delta windings and the mill head star!

But the real question is what the winding config is on your new motor. Is there a connection cover you can remove and get a photo of it? If it can be easily converted to delta then the neutral, the place where all the windings are connected at one end at the centre, would be accessible at screw terminals and/or have links.

The rating plate says it's 0.45kW 2-pole (2800 rpm) and 0.3kW 4-pole (1400 rpm) so a VFD rated at 750W (0.75kW) should be fine.

Fitting a VFD it is best to scrap all the switchgear between the VFD and the motor, you shouldn't be tempted to switch the motor when running with an existing contactor, it will instantly blow up the devices in the VFD. All the control can be done by the VFD - start/stop, speed, reverse, soft start, e-stop etc. Just need an isolating switch and filter where the mains comes in, and a control pendant.

I've fitted 3-phase to a couple of mills using standard delta motors with ordinary VFDs and also fitted a Newton-Tesla pre-packaged unit to my Super 7 - but I haven't any direct experience of 2-speed motors. It can't be that hard though. With a VFD I doubt you would want the 2-speed feature anyway as the VFD does it for you.

I have heard that these people are very helpful: https://inverterdrive.com/