Hi Rod,
That one looks involute, I have an old banjo mandolin where the cuts look more like splits cut in the worm wheel!
Also odd ones on this Walthari Mittenwald mandolin, mid 20th century. A very skinny thread, they appear gashed as triangular tooth-spaces then hobbed or just worn to a slightly enveloping profile for the very skinny worms whic appear to have been cut with a round-nosed tool.

All my modern instruments (that use this type of tuner) have what appear to be involute gears cut to match the helix angle of the worm. They generally appear to be hobbed and so partially enveloping the worm. I'm sure my oldr instruments all have less 'engineered' arrangements.
I'm sure pretty much anything will work, but a well made involute-based worm and wheel is probably going to give the smoothest action.
Neil