Others probably know more than I do, but I would guess that the glass has been replaced, perhaps as a result of breakage. It is difficult for me to imagine a careful worker or designer using plain nails to hold in place a large piece of glass. What I think is much more likely is that either putty was used (you should probably be able to find remnants of that), as is often the case in window glass panes, or that a flexible strip of wood or some other material was formed around the inside of the glass, then secured in place by nails. Such nails would not have been finishing nails, but nails with heads on them.
Using nails – even finishing nails – to directly hold in glass seems to me like bad practice. Also, the edge of your glass, especially on the right of the photo, shows evidence that it was cut and fitted by a somewhat inexperienced worker. A nice piece of flexible wood strip or of caulk would conceal those irregularities.
Accordingly, I wouldn't feel comfortable using the nails as a gauge of age. Pictures of the works might elicit better ideas. Meanwhile, it is very good to hear that your clock is such a good timekeeper.
Is the wood solid or veneer? Older clocks almost always used veneer to save money. What is the size of the winding squares? They look rather large. How long does the clock run between windings?