This one is a brass like for putting a Balance wheel in it.

It has no marks on it.

I think these where made at a watchmakers school by a pupil.

There are 4 disc that are removable. 2 are sitting next to the base. 2 are on the base.

I'm guessing the second one is a kind of clamshell hand puller.

They all belong in a Museum, as for use . . . Just "Guessin" . . .

lol @ dave

David, I take seriously everything you comment on here. But, in this case I think I could've come up with something just as good.
ha

Dave

The thing you are calling a pivot polisher. looks more like a balance wheel straightener.
But it does not have a arm that you put close to the wheel to make sure it is straight.

The item you say is for balance staff remover. I think it might be for roller table remover.

Because the indented area is convexed.

EXACTLY!

Ouch!!