An ambitious project I have found lathes are an order of magnitude more difficult to upgrade than milling machines, but I have only done one lathe.
There are a number of books/web pages on how servo systems work. You might take a look at this page for a starting point White Papers | Galil
There is no standard nomenclature for the drive/controller I/O, each manufacturer has there own naming conventions, however, there are things that you expect to see that are common to all.
On the drive (amplifier) end you will find input(s) for a control signal(s). This could be analog, step & direction, PWM, or direct digital control, also at the very least there will normally be an enable input. Other inputs might include reverse rotation inhibit, run and stop signals, and other stuff. Then you might find encoder outputs, alarm outputs, and other various outputs.
On the controller end you would find the mating inputs and outputs for the above signals. Typically the analog control signal is +/-10V, the other signals may be 5 to 24V depending on the controller/drive. If your controller and drive signal voltages are not compatible, then installing relays between them is required. This is normally done with optocoupled rather than electromechanical relays.
In your case, you actually have good documentation for both the controller and the drive so it is a matter of translating the signal names from one to the other. The drives take a +/-10V analog input and it looks like the Oak board will output that signal, There is an encoder output from the drive, this would connect to the encoder input on the controller. These items and an enable signal to the drive should get the wheels turning.
One note of caution: Mechanically disconnect the servo motors from the lead screws until you know you have the servo motor under complete control. A runaway servo motor can do some serious mechanical damage to the hardware.