I have been following this thread, own a Camelback drill and have been around a number of them. If the shaft is the horizontal shaft at the top of the drill press, it is usually a fairly simple shaft in terms of machine work required. My own take on things is to try to avoid the matter of doing a weld repair on the shaft. As noted, you will have to turn the shaft undersize to a true and constant "cylinder". The worn areas that are turned undersized will need to have the shoulders transition back to full/original shaft diameter with a radius or smooth taper to avoid a bad stress concentration. The welding will likely be several passes deep. The potential, even with MIG welding, to warp the shaft is quite good. How good a quality weld you get with MIG is another matter.
As an engineer and Certified Welding Inspector with many years experience, I tend to shy away from MIG welding on anything that requires a sound weld with good penetration. On new work such as plate or light structural, GMAW (MIG) is OK, but we tend to shy away from it on piping, prerssure vessel and heavy structural work. We also avoid GMAW for something like building up a smaller diameter worn shaft. Unless the GMAW is put on automatically, such as in a lathe with a closely controlled process, there is too much chance of warpage. We normally would use GTAW (TIG) for a job on a smaller shaft such as you are describing, if the shaft were something special, such as a pump shaft having a stainless steel overlay. Otherwise, we just replace the shaft. A small shaft with no special features to it is not worth repairing with welding.
For the shaft in a camelback drill, I'd get a piece of "Stressproof" turned/ground/polished shafting steel, turn to whatever other diameters are needed, and mill the keyways. Stressproof is wonderful steel, about .40% carbon IIRC, and will maintain dimensional stability if you machine it. Cold rolled will often warp like a snake if you machine it assymetrically, and cold rolled does not have the strength of Stressproof. A lot of the older shafting was made from 1030 steel (.30% carbon), and had a bit more strength than cold rolled. For a few bucks, a piece of TGP Stressproof will save you a LOT of grief and save you a LOT of time. I've made a number of replacement shafts for various machinery out of Stressproof TGP, doing jobs in my home machine shop as well as incorporating into designs at work.
If you have a hole in a pulley that is worn egg shaped, I am assuming this is on a tight/loose pulley (clutch) arrangement ? A pulley which is "tight" (keyed to the shaft) should not have an egg shaped bore. A "loose pulley" is meant to freewheel on the shaft and the belt is shifted onto it to disengage the power from driving the drill press. Lineshaft driven drills often had the tight and loose pulleys on the bottom horizontal shaft with the bottom cone pulley. If the bore in this type pulley is worn egg shaped, then it needs to be bored oversized and a bearing sleeve shrunk in and bored to correct size for the shaft.