Way back in the old days people used to resharpen mower reels using an ordinary toolroom lathe and a high speed steel tool, Graham, but that was backyard practice, not something a greenkeeper would do. At best there will be burrs to remove, and you would have to use a large nose radius tool, a fine feed, and extremely light cuts (or you will deform the blades where they attach to the frame of the reel, and along their length). An excellent job can be done with a toolroom lathe using a toolpost grinder, provided it has a large grinding wheel - say, at least 4" diameter. The limitation on using a more conventional small toolpost grinder is that the wheel will wear down during each cut, and the diameter of the reel will therefore vary slightly from one end to the other. This is not disastrous, because you can adjust the clearance of the two ends of either the reel or the bedknife (depending which model mower) independently.
I think the main reason to have the job done by a specialist, using a special purpose reel grinder, is that the chances are nothing will go wrong and you aren't risking using up a lot of the "meat" on your blades through having to repeat the operation a few times before you get it right.
You can sharpen the bedknife with a surface grinder, but to get it flat along its length it needs to be attached to its holder when you do it, and supported at the two ends, as it is in service. This means using very fine cuts, or it will deflect downward when grinding near the center. I am not clear on how this was done in the old days - it may have been usual to just replace the hardened steel bedknife, then lap it flat after attaching it to its holder. Remember, the quality of the final mowing job depends on a straight, parallel reel, a straight, flat bedknife, and a straight front edge of the bedknife that is precisely parallel to the axis of the reel. No burrs on the reel blades, of course.
If your mower has had its blade adjusted many times you are likely to find that the adjusting screws have their threads stretched out of pitch at the precise point where they are usually adjusted. This makes adjustment just about impossible. (If you have ever adjusted the tappets on a very old car that has suffered from ham-fisted mechanics over the years, you will be familiar with the problem.) I suggest you inspect the adjusting screws carefully, and replace them if necessary.
