I did a similar repair on my old Drummond lathe some years ago. Used the spool-shaped sleeve and soft soldered it in position. I tinned the half nut body and the half-spool piece with a thin layer of soft solder then put them together and heated until I could melt a bit more solder into the join for good measure. Has been working well for me for years since.
Between the large surface area of the soldered join and the definite location of the shoulder on the spool, it is plenty strong enough. If you've ever used a leadscrew with a hand wheel on the end as the old Drummonds and some Myfords do, you'll know the forces are not great. You turn the leadscrew handwheel with light finger pressure. Screwcutting would be a bit more pressure but light cuts so still not great.