In my years as a clockmaker, I put many a bushing into many a clock. Now I find that similar techniques are necessary when repairing watches.
Watches with less than 17 jewels have holes in the brass plates which act as bearings. With time, these can wear oval, which mis-meshes the gears and causes the watch to stop. The only cure is to insert a bushing, as shown on the left.
Clock bushings -- which also come in tiny sizes -- work quite well in watches. The bushing on the right has a .10 mm bore, which can fit very small train wheel pivots. They are also available in .15, .20, .25, .30 millimeter bore sizes, and up. Clock bushings are available to cover all watch pivot sizes including the center wheel and even spring barrel pivots. The outside dimension of this one is 1.2 mm.
I ream the plates with the available clock bushing reamers which come with most reamer sets. This one, shown on the left, reams a 1.2 mm hole which will fit the bushing above. Before using the reamer, however, I enlarge the hole in the watch plate with a #39 drill, which is a little smaller than the bushing reamer. I use a drill press for the reaming operation to assure reaming vertically.
When bushing clocks, I'll normally broach and file the worn pivot hole until it's round and centered before reaming. You can do this here, too, but it's often not necessary.
The bushings are too long for watch plates, so I put the bushing in a number 12 collet in the lathe, shown on the right, to shorten it with a graver. You can skewer the little bushing with one of the little broaches shown below to avoid losing it when inserting it into the collet.
I put the bushing in the hole in the plate with a staking set. These bushings are .01 mm oversize the hole reamed, which is a snug fit for a watch.
Once the bushing is in, its hole can be broached to adjust it to the correct pivot size. Broaches are available in tiny sizes, as shown here. These came from TimeSavers, which also have reamer sets and the bushings themselves.
Bushings can also be shortened and made flush with the inside of the plate using homemade pilot cutters. The ones shown here are very small -- the one one the right has a pilot diameter of only .008" inch, which will do for even fairly small pivot holes.
Details for making pilot cutters are found here.