Those look like standard spools used for all kinds of wire. I have bunches of them at the plant. People don't throw stuff like that away because the boss may find a use for them.

I have several long extension cords wound on them, as well as excess air hose.

See if there is an electrical supply place that will give you some, or suggest a customer that buys from them and might let you have some.

An arbor for your lathe might be a simple as a threaded rod with a couple of wood/aluminum hubs for the ends of the spool. I am sure you could come up with a fixture to hold the broken spool and let the wire out to rewind on the new spool.

Watch out for crossed up spots as the wire comes off the original spool. Have to stop the lathe fast, or leave the take up spool rather loose, so it will slip when tension is too great.

Paul