Here is a clock plate showing two very worn pivot holes and one of the wheels to show how much of the pivot has worn away. Here it is held in a collet in the lathe.
Clearly this pivot cannot be reused as it will be far too frail when ground and polished back to parallel.
The answer is to replace the pivot. The first step is to cut off the pivot and grind back to the shoulder.
Next mark the centre of the arbor using a carbide centre drill in the lathe tailstock. Never use a twist drill as they always wander and lose the centre.
Drill out the new pivot hole using a flat or spade drill of the right size and keep oiled. I make these flat drills using solid carbide rod and a diamond lap.
The new pivot should be a drive in fit and can then be cut or ground to length and polished, first with a diamond file and then with a pivot file using oil as a lubricant.
Mainsprings get dirty, sticky and gummed up over a period of time as old oil dries up or deteriorates. Often they can be cleaned and reused. I made this tool to make cleaning a mainspring easy and safe. It has been in use for more than twenty years (and yes, it looks like it!)
This tool is made to fit a slide on the office desk I use for my clock bench but there is no reason why it cannot be fitted to a piece of ply and clamped to a bench.
The mainspring is hooked over the bar, stretched back with one hand and then scrubbed with coarse steel wool in the other hand. Turn over the mainspring and do the other side.
I made this tool using two pieces of angle iron with the rod attached by drilling and welding. Four bolts attach the tool to the slide.