How much hard work do you realistically expect the lathe to do?
It's in the same bag as an adept so the headstock will be worn as well.
Use a decent steel but not one that will be hard to machine to a good finish and you can probably do as good a job as the original.
The original Adept spec was 'mild steel' (probably of good engineering quality
).
Bear in mind that if the steel is too tough, you will struggle to put a morse taper socket in it with a reamer.
If doing the adept spindle again I would turn it all in one go, starting with the nose, then boring the socket and finishing everything else with a running centre in the socket, but 101 ways to skin this cat.