I found a 1940s craftsman(atlas) drill press a few years ago and rebuilt it.  I removed the bearings, pressure washed the castings, etc.  
in the process of removing the bearings from the quill I used a brass hammer while holding the parts in my hand as they were a bit tight.  Upon reassembly I discovered that the steel is extremely soft and I had mushroomed the end of the taper where the chuck attaches.  I put it back together, turned the drill on the highest speed, and used a fine file to remove the mushroomed tip from the taper.  The chuck went back on and all was fine for a couple years.  

The chuck came off while drilling a few months ago and now it refuses to stay on.  The taper on the shaft is visibly not smooth and shiny and needs to be restored.  I’ve gotten it as good as I can with fine Emory cloth and a hard backer.  I don’t have a lathe to attempt to get a perfect angle on it.  

There are two styles of chucks used in these machines, the early style which I have attaches with a taper only.  Ie. You can’t put any side load on the chuck or it will come loose.  The later style have a threaded collar that holds the chuck onto the taper so you can have side loads on the chuck.  See picture below, my chuck is like the far left one.  Other picture is actually my chuck.  

There are used quill shafts and chucks with the collar attachment on eBay, but I’m fearful of the condition such an old part might be in.  (Ie. Warped shaft or abused chuck making it not run true).  

What would you do?  I’m leaning towards gambling on a used chuck with the flange because if the chuck is trash I can put the insides of my chuck into the new chuck body that has the flange.  

I have a bunch of old and not so old Jacobs chucks, but none that I would call great.  My Christmas list included a couple of Shars keyless chucks, and I may never use a Jacobs chuck again.  I got a 1/2", and 1 3/4".

https://www.amazon.com/dp/B09LJ4JGBK/?coliid=I2UKRQP474MG1A&colid=1GXYJSC0UR632&psc=1&ref_=list_c_wl_lv_ov_lig_dp_it

ETA: I realize that this doesn't answer your question.  Looking at those chucks reminded me of my old ones, and made me feel bad for you.  

My chuck is mint, it’s the quill shaft that is the problem.  New chuck would only be to get the flange that holds the chuck onto the taper.

You could try lapping it.

Lapping is the way to mate tapers.  For a taper in that size and condition, I'd start with 320 grit, and finish with 500 grit.  

Set the drill as slow as it will go, apply the Clover lapping paste to the shaft (oil based is better for that purpose), and gently press and hold the chuck into the taper. Back out frequently, and re-distribute the paste.  Apply new paste only as required.  

Once 85% to 90% contact area looks grey, then clean off all the grit and grease.  Then repeat with the finer grit.

I had some 220 lapping compound and tried it using a piece of stainless as the lapping surface.  Made the taper surface a matte finish, need a finer grit.  But the surface is uniform now.  
I drilled a few holes no problem, previously it had started coming lose Almost constantly.   With a decent size drill bit In the chuck I can even wiggle the chuck back and forth and it doesn’t come off!  

I had tried fine emory cloth before, but it never left a very uniform surface for some reason.  I’ll get some lapping compound around 500-600 and it should only get better I hope.  Thanks!

Pretty sure you would use the chuck, not the stainless, to lap.  That way you are matching up the contact surfaces.

This.  Lap the two mating surfaces.

I considered that but didn't want to risk screwing up the chuck.  
So far so good, I'll try the finer grit with a flat surface first.  If the chuck still comes off in use I'll try lapping the chuck and quill together.  

I don't think you'll hurt the chuck.  All you're doing is removing the high spots.  You won't change the taper.  It's actually SOP.

So you want to lap a round surface with something flat?

The point of lapping is to fit two pieces together by removing the slight imperfections, and in this case removing any slight variance in the tapers.