Michael’s table showing DIMENSION OF CHUCKS OF VARIOUS MANUFACTURERS illustrates an awkward problem, which is that Watchmakers Lathes aren’t standardised!
Less of an issue when the lathes were new, because customers knew what they were buying. And the lathe came with a set of accessories, often seriously expensive:

Back then repairing mechanical clocks and watches was big business, so large numbers of lathes were sold. Many not particularly well-made or supported, others excellent. Unbranded lathes seem quite common.
As time passed, clock repairing as a business went into decline, so many lathes became unloved, badly stored, separated from their accessories, or scrapped. Decades later, we pick up the survivors, and finding out what it is can be difficult. Not even obvious how they were used.
I think Shaun will have to carefully measure his head and tailstock. With luck they match one of the well-known makers.
Though I came close to taking up clock-making as a hobby and swotted up, I don’t have a feel for the minimum number of accessories needed to get going. Building a clock, I guess much could be done with a few collets and a shellacked faceplate. Repairing old clocks for money might need a lot more because shapes and sizes vary so much. Can a clockmaker advise please – what else does Shaun need to get his lathe operational?
As an aside, I’ve watched many episodes of “Repair Shop” whilst caring for mum. They have a Myford in the background, which is never used! In contrast, a watch-makers lathe gets lots of screen time. Unfortunately they don’t explain or show the accessories. Might all be done with a one graver and a single collet, but I suspect there’s a drawerful of collets and other goodies off-screen. TV isn’t good at techie subjects.
Dave