My first lathe bought when I was about 13 was one of those but painted blue, a B&D and I did the same as DTR eventually and converted it. I kept the bed and tailstock, bought a couple of plummer blocks and a shaft which I drilled and tapped to take the drive and faceplate and threaded the outside for a small chuck. Mounted the whole thing to a 1/2" steel baseplate which I could barely lift.
The original drive etc weren't fitted into a chuck, you just removed that and screwed then straight into the end of the drill (something like 3/8 unf rings a bell). I'll have a look tomorrow just in case the drive is still there - probably not but you never know.
Bob