I don't know if you have looked at Ivan Law's book Gears and Gear Cutting (Special Interest Model Books, Workshop Practice Series, No.17).
He includes and explains the basic, most important geometry from first principles; whereas full treatises on gear theory dive straight to oceanic depths of complexity; and generally, considerably de-mystifies the subject of designing and making spur, bevel and worm-gears.
One chapter and shows how to make gear-cutters, both single-point (fly cutters) and multi-tooth. Now, this is for involute teeth and I do not know if your clock uses that or hypocycloidal forms – or whether it matters in the light of others' comments here.
(I think, but am not sure, that hypocycloidal-tooth wheels are the usual in small clocks, as the shape loses a little strength but more importantly reduces friction, compared to involute shape. You can't mix them though, in any application – excepting the lantern-pinion oft found in clocks, gears of any given pitch, tooth-form and pressure-angle will mesh only like-for-like.)
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As a digression from the lantern-pinion with its rod-teeth, for anyone modelling certain types of very old equipment; some such as manually-operated cranes and horse-gins, made cheaply for rough work, used a very crude tooth-form commonly called hollows and rounds. The tooth tips and roots were hemi-cylinders meeting tangentially on the pitch-circle, simple to make but incapable of transmitting much power and then only at low speed, with considerable friction and jarring. That was not from cost and difficulty in precisely cutting metal to cycloidal shapes, in their day. Instead, despite the illusion given by the shape, hollows-and-rounds teeth do not roll on each other as involute and related teeth do, but slide with changing velocity ratio.
The Classical Greek orrery, the Antikythera appears from the photos, to have triangular teeth – but that beautiful instrument still worked!