I think it would be difficult to make a double ended spindle for a lathe that is not built for it. This because ball bearings are usually pressed in place from the outboard end which is often thinner than the threaded end of the spindle. If your lathe spindle is thicker at the bearings than it is at the business end or if it has corrugated babbit bearings this would be no problem but I doubt it is.
I know several machinists locally who could make a lathe spindle. There are special reamers for morse tapers in soft iron and I doubt that a spindle taper on a wood lathe would have to be hardened.
I think there has to be machinists who can do this somewhere in the countryside in your country too. Look for a farm equipment or heavy machinery repair shop and walk in and ask if they know a good machinist who does one off jobs. If there is one in the area they may know him. Spare parts for farm equipment and heavy construction equipment are often expensive and hard to get on short notice so it is fairly common in that field to bring the broken part to a machinist to be used as a pattern for a new part.
Left hand threads are notoriusly difficult to turn so expect to pay a bit for the service.
Good luck!
