It is not unheard of for manufacturers to use the same diameter leadscrew on both their metric and imperial machines.
It makes manufacturing and stock-keeping a lot easier.
So it is possible to have 7/8" diameter with a 3mm pitch. Similarly, it is conceivable to have 22mm diameter with 8tpi cut on it.
The translation gear refers to the special gear or pair of gears that allow imperial threads to be cut on a metric machine or metric threads on an imperial machine.
Whichever are the native units of the machine, the geartrains for threads in those units will be simple (e.g. 20tpi on an 8tpi leadscrew needs something that simplifies to 5/2). For threads in the non-native units, the geartrains are more complicated and the numbers do not reduce to simple fractions.
Once you reassemble it, you can confirm the pitch of the leadscrew by measuring how much the carriage moves for say 16 revolutions of the leadscrew. It does not need sophisticated measuring equipment for this: a steel rule will distinguish between 48mm and 50.8mm and a chalk mark will index the rotation angle.
In case you did not find it, the actual manual for the machine is here:
https://www.manualslib.com/manual/1906935/Chester-Craftsman.html#manual
It worries me a little that you say the dials are in both units. If the exploded diagram in the manual is correct, then one set of dial graduations will, at best, be very difficult to use.
Dual reading dials that are easily-usable in both sets of units are available (best known is Gamet dials on Colchesters) but require a few more parts than are shown in the manual.