I do not recommend you use the E-Stop and safety interlocks to kill power to the VFD, if you hit the E-Stop the VFD will often free wheel to a stop, although some can use regenerative power to do a controlled braking. Typically the E-Stop and safety switched are all serially connected to the VFD low voltage input signaling so, would kill power to the run commands. You do not want the forward/reverse switches directly wired the respective VFD low voltage input because the lathe can start if you release the E-Stop or any of the interlocks. Typically what is done is to use two latching relays to for the forward/reverse commands, these are typically 12 or 24VDC. Alternative is to strip out the high voltage to the contactors and use them to switch the low voltage VFD signals. These requires replacing the contactors as the contact resistance is an issue with used ones.
Unclear on what you want the foot brake to do, if you want it to stop the lathe with braking then it just breaks the signaling pathway, but you still need some form of latching relay design so the machine does not restart when you release the brake. I have also designed systems that use momentary buttons for for/stop/rev which some people have preferred. If you have a manual brake, then you want to wire the stop switch to issue a free run command to the VFD so the VFD braking is not over riding the foot brake. You can also program the VFD for different braking rates.
Attached is a control system I designed for the G4003G which uses the WJ200 VFD, the jog circuit would vary based on the VFD. This design uses 3 pole relays, the 3rd pole could be wired to operate a coolant circuit, otherwise 2 pole relay can be used. The relays should use the snap in protection diodes that fit into the relay base to prevent reverse voltage spikes when the relays open.