Hello there,
I've only seen one real life toroid winding machine in my life. It's quite a complicated machine.
It winds the turns around the core as it rotates the core around, and the bumpers that hold the core are cushioned and they turn to make the core rotate. It's quite amazing to watch, maybe there is a video somewhere on the web.
To do it by hand, a 'shuttle' is used. First the total length of the wire is computed, knowing the core surface circumference and taking into account how that changes as the turns are added. This length of wire is then wound onto a small bobbin that can fit through the center of the toroid. The turns are then applied by pushing the bobbin through the center of the core repeatedly until the correct number of turns are applied.
A layer of tape would then be applied, then the next winding.
Bifilar windings are done the same way except two wire lengths are cut and wound onto the bobbin.
The bobbin is usually referred to as the "shuttle".
For sine waves the formula for the flux density due to the primary winding is:
B=E*10^8/(4.44*F*A*N)
and for square waves it is:
B=E*10^8/(4*F*A*N)
which is the same as for EI laminated cores.
Here B is flux density in Gauss, E is RMS voltage in volts, F is frequency in Hertz, A is core cross sectional area in square centimeters, N is the number of turns. Solve that for N to get the number of turns.