Is it possible to change the handle and how?
Yes, you can replace the handle. But there's a decent chance you can repair the existing one strongly enough that the repair will last for many years if you'd prefer to do that.
This doesn't require a lathe but obviously if you want the new handle to be round then it would be the most efficient way of forming a replacement. A turned handle can also be formed using a drill as a rudimentary lathe. But a perfectly round handle can be shaped entirely by hand processes, using some combination of plane, spokeshave and rasp/file work, followed by scraping and/or sanding.
The handle doesn't have to be round1.
Regardless of whether the handle is turned or hand-shaped it has to eventually be two pieces so that it can be glued around the shaft. Generally you want to start with a single piece of wood, although it's not mandatory it gives you nearly contiguous grain. The handle can be fully shaped and then carefully split, or roughed into a blank, sawn in two and then the two pieces temporarily joined together2 before shaping is completed. Obviously the second method can be used if two separate pieces are used for whatever reason.
Fix the existing handle
Because the break looks old, and is anyway on something that sees regular hand contact it's not as straightforward as dribbling a little glue into the crack and clamping it closed — for a start because only fresh wood surfaces glue reliably. But add in any contamination from oils from the hands plus shed skin cells, or really any sort of dirt, and those will prevent any adhesive from getting a solid purchase on the wood.
There's an additional possible complication, cracks like this sometimes aren't fully closable. Through a combination of factors the wood may have shrunk to the point that the inside diameter is now too small to close on the shaft of the handle.
So for this reason stabilising the crack by filling it may be the best route. I haven't done this myself on the sweep handle of a brace but a friend did such a repair recently so I've seen one in the flesh and it worked perfectly. It is early days yet (especially given the multi-generation lifespan of such a tool) but some months on it is holding up nicely.
First step in either case is to clean the interior surfaces of the crack starting with gentle scraping of any obvious gunk, using a dental pick maybe, followed by thorough solvent cleaning. I'd start with mineral spirits (UK: white spirit) and then use either denatured alcohol (UK: meths) or acetone. You can scrub with any stiff-bristled brush but one of the best tools for this sort of thing is a toothbrush because the nylon bristles are particularly stiff.
Note: you can gauge how effective your cleaning has been by placing a drop of water anywhere along the split. If the water soaks straight into the wood you're good to go. If it beads up and soaks in slowly further cleaning would be advisable.
Now mix up some epoxy with sanding dust to a consistency somewhat like mayo, spread it over the crack (overfill slightly) and leave it to harden3. Then file, scrape or sand the fill flush with the wood. Follow this by scraping or sanding both handles as much as you care to and finally apply your finish of choice.
Obviously there's a chance the epoxy will stick the wood to the shaft but don't worry about that. My friend managed to carefully work some wax into the gap without getting it on the crack in the wood, but there's no need to go to the trouble. For a start on an old brace the metal is probably pretty filthy to begin with, making a good bond highly unlikely ^_^ But if by chance it does stick firmly enough that the handle won't spin heating the metal using a heat gun or hairdryer is sufficient to cause epoxy's grip to weaken.
How did they put the handle there in the first place?
I wondered the same thing and it's something that has perplexed many a brace owner! The answer is very simple once you know it — the wooden handle was slid into place before the brace was fully bent to shape. From what I've read generally the first bend (near the chuck) was done first, the handle slid into position and then the top bends done.
1You could go with hexagonal or octagonal if you wanted to try those out. Either cross-section could be further refined to round if you find you don't like how the polygonal handle feels in use.
2Traditionally this was often done using a 'paper joint' which works just as well today as it always has, but these days superglue and tape is an option, see How do I temporarily attach two pieces of wood together for machining?
3It is best to wait for it to fully cure before proceeding. Even with 5-minute epoxy I wait 2-3 hours minimum even in the summer, but try to forget about it so it's at least half a day before I get back to it. With 1-hour epoxy I would wait at least six hours for it to set, overnight preferably. In case you don't know epoxy hardens via a chemical reaction and like all reactions it is temperature-sensitive, so in colder weather you need to wait longer than when it's warm.