Making provision for oil lubrication, (rather than grease ) sounds to be an excellent idea.
Don't know the lathe, (Lathes UK illustrations seem to show a pin in the casting at the "chuck" end of the casting, and an oil cup at the rear end of the casting..The smaller gear also appears to be pinned to the shaft.) Longer study may provide some clues. Back Gear looks to be engaged by rotating the lever at the back end of the shaft, which suggests that there is an eccentric involved somewhere.
Wild guessing; but it could be that the shaft is located endwise by the pin at the chuck end, engaging in a groove, whilst the gears are run on a journal which is eccentric to the shaft. In this way, rotating the shaft with the small lever, will throw the Back Gear cluster in and out of mesh. But there must also be some means of clamping the drive pulleys to the mandrel, which will need to be disengaged when back Gear is being used., to prevent everything locking solid.
Once the end locating pin has been removed, and the pin from the small gear, it ought to be possible to remove the shaft (Might need to be gently driven out )
The lathe can then be used to drill and tap the chuck end of the shaft for the oil nipple and oilway, before being cross drilled;
(IF the oil cup at the lever end does not provide this function. Possibly by a groove under the oil cup leading to a cross drilling, a longitudinal drilling, and cross drillings into the journal(s )
Once clean, it can be reinstalled and you are away!
HTH
Howard