Tuthill Pumps is still in business. They still offer pumps for "built in" applications such as machine tool lubrication systems. Have you contacted Tuthill Pumps ?
The other option is to contact LeBlond, Ltd. They maintain records on LeBlond lathes going back to the 30's or 40's, and have some parts available. Since the Tuthill Pump is an "outside purchased" item, they may be able to get the rebuild kit. I know the bigger LeBlond HD lathes had a "Cuno" strainer built right into the side of the headstock. This is a "stacked plate" or "porous edge" type of strainer. On this type of strainer, you turned a "tee handle" to clean the strainer. This rotates the discs of "stacked plates" against a scraper blade. Eventually, the particulates and sludge removed has to be cleaned out of the strainer. Typically, the bolted cover plate with the tee handle is removed from the side of the headstock as a unit. The strainer assembly is NOT disassembled beyond that point, but is washed in solvent and blown clean, and the accumulated sludge in the strainer body ( a cavity in the headstock casting) is cleaned out. This Cuno strainer is typically on the discharge side of the oil pump.
On the suction side, there may also be a "basket" type of strainer on the suction side of the pump. If this strainer is partially clogged, it would cause the gear pump to have a hard time "picking up". Typically, the gear pump is mounted low to the rear of the headstock. I'd start with the simplest: clean the strainer(s) and then check the suction piping to the pump for sludging or obstructions.
The Tuthill pumps are usually an "internal gear type", and are pretty well bulletproof. I'd go for the simplest things first, like a partially clogged suction strainer or suction piping. Another diagnostic tool is to put a vacuum gauge teed into the suction line between the pump and the suction line connection to the headstock- if possible. If you start the lathe and the vacuum gauge hardly registers, then the pump is worn. I'm not sure what a "normal" vacuum gauge reading should be, but I am guessing 8"-10" of mercury would be reasonable for a healthy pump. If the pump is set low behind the headstock, the oil is supplied under what is known as "positive suction head". This means the pump does not have to "pick up" prime as the positive head on the oil keeps the pump filled. If the pump is weak or not delivering and is mounted so it has "positive suction head", I'd start looking at the suction piping between the headstock and the pump and look deep down inside the headstock for a clogged suction intake.