First of all, there is no "power" difference between 240 and 120 volt motors. The difference is with how much current it draws from the line. Basically, how big a fuse would have to be. Otherwise, a 1/2 HP motor is a 1/2 HP motor. The only difference would be the cord connector type and fuse size. I happen to like 240 volt machines, but my lathes are 120 volts because that's where the cables run.
As to the size, or HP rating, I have a 101.27440. A little older than your's, but essentially the same machine on an open frame. I am underpowered with a 1/3 HP motor. But, I'm a model builder and don't push the machine hard. That small would be your call. A 1/2 HP is useable. If you had the machine in commercial service, I would suggest 3/4 HP. For that particular machine, any thing over that would be overkill. It would cut fine, but there are a lot of sleeve bearings that would wear out faster with more power.
Now, I wouldn't want to arbitrarily hook it to 240 volts without triple checking the wiring connections. As noted above, you'll have to run down the schematic for that specific motor. It will be one of many wiring types from the same source. Motor speed is based on line frequency rather than voltage. With that type of motor. There are motors where speed is a function of voltage, but that's not applicable here.
I would look more toward the starting circuit. First try the spindle floating, pull the lever out to disable the final belt reduction. Quickly start the motor and spin the large reduction pulley by hand. Watch out for your fingers. The motor should come up to speed. If it doesn't, shut it down quickly and start looking for a mechanical problem. If it does come up to speed, check the start circuit / capacitor.
You stated the reversing switch was disabled. That indicates to me that there is likely a problem with the start circuit and the PO tried to "fix" it but couldn't. It may well be why the machine came up for sale. It's easy to troubleshoot remotely... ... But that will give a starting point. Start there before spending a lot of money on parts that may not be needed.
Bill Hudson