I have a small metal lathe, maybe I'll try a fix, and if it turns out poorly, I'll just check it up to experience.
It's simple work, but it has to be done with precision and care. Don't be half mast about it.
Is your lathe large enough that the shaft will fit through the spindle, so only the end protrudes? To do good concentric work the shaft ought to have support at both ends (spindle nose where the joint will be worked on, and outboard of the spindle where the business end sticks through); and a method to dial both ends in with a DTI (Indicator). Repairs and joint work are not really 3 jaw work, a 4J or an adjus-tru chuck, or a good collet system are essential. With careful work, onesize/one use wood or plastic cones can be made for the outboard end instead of a chuck.
If you just want a cue fixed don't mess with it.
If the work interests you and you already have metal working equipment, the task is basic. It will be time consuming the first time. Turn off the old collar leaving a parallel round tenon. If the old collar is short, you can make the tenon longer, so the funnel shape is less extreme. On one of your other lathes,:wink: bore a phenolic collar to fit, leaving the OD oversize. Glue it on, using the lathe tailstock ram as a pusher/clamp to hold it from moving back out when you are not looking. Don't over-tighten or it will push the shaft back & you will have to go through all the trouble to re-center it when turning the OD.
When the glue sets, turn the collar down to the mating OD. Then swivel the lathe compound a little and cut the necessary taper to blend by means of the lathe compound. Feed by hand, because the taper will be steeper than the shaft taper and you don't want to create even the least undercut or score. Blend in the last bit with sandpaper, and keep the sandpaper off the wood. Maybe wrap the wood with a couple layers of masking tape before starting to sand. Using a wood backer with stick-on sandpaper will give good control. Don't forget to face the joint while it is still set up running dead nuts concentric.
The weird thing is, if you beat all your opponents at the local watering hole, or win a local tournament, more than a few people will want that precision "funnel joint mod" done to their cues, too.

If the logo is still visible on the cue, some will be calling the factory trying to find out how to get that special shaft.
Other notes: People who make or work on cues eventually get a collection of (usually plastic) sleeves to keep chuck jaws from marring finished or near finished cue surfaces. For a one-off the part can be wrapped with a few layers of heavy paper, taped. This assumes a method to center the work in the chuck, either 4 jaw independent, or an adjust-tru of 3 to six jaws, as you prefer. It also assumes you don't have a touch on the chuck key that will crush the work regardless.

Without chucks that can be dialed in to change center location, it is probably not a good idea to attack joint work. though there are ways around it if you are already a decent machinist.
smt