[old_tv_nut]

I presume you are planning to use an existing yoke? That would be a good idea, not only for image geometry and convergence, but because you could measure the yoke currents and drive voltages that you need to produce in an existing set. Then you "only" have the problem of winding transformers that will match a tube circuit to a yoke that was designed for transistor drives without the nearly impossible job of determining yoke design.

Some things to know:

Vertical sweep operates the yoke as mostly resistive, with some inductance. In other words, you need a class A (or close to it) drive, and will have a voltage waveform that consists of a sawtooth plus some amplitude of pulse during retrace.

Horizontal sweep operates the yoke as mainly inductive, with a switch-mode horizontal output that applies an essentially constant voltage to the yoke inductance, producing a sawtooth current (in concert with the damper tube), and a large voltage pulse during retrace. The inductance producing the sawtooth is the parallel of the flyback transformer and the yoke. Impedance matching is obtained by the winding ratio of the taps for the horizontal output device and the yoke. To use a tube HO to drive a low impedance (transistor) yoke, you will need a much greater ratio than was used in the transistor set the yoke came from. This means the tube capabilities will be matched (higher voltage retrace pulse and and lower current during active trace compared to a transistor). In tube circuits the damper tube may need a separate winding ratio of its own, whereas transistor circuits often put the damper on the same tap as the output transistor. Transistor circuits might have HO collector retrace pulse on the order of 1 kv or less, while tube circuits might have multiples of 1 kv. Retrace voltage and retrace time are also affected by the value of the retrace tuning capacitor.

Service info typically says "do not measure" and doesn't tell you the amplitude of the pulse, which takes a special high voltage scope probe to measure, so they didn't want to tempt technicians to blow out their scopes.

By the time sets were in wide commercial production, no designer in their right mind would take the trouble to go back to basic theory to calculate winding counts, inductances and resistances unless forced to by radical changes in design, like going to transistor circuits. Instead, they woud typically take a similar design and modify it slightly to match the new needs (the new yoke impedance is x % lower, so change the winding ratio by that much) and then fine tune the prototypes on the bench.

The flyback transformer is also responsible for generating the high voltage. For least high voltage variation with beam current, you actually want the HV winding to resonate at three times (or even five times) the primary retrace frequency, to present as flat-topped a pulse to the HV rectifier as possible. This is a refinement that I wouldn't expect any DIYer to make, as it depends on the distributed capacitance of the HV winding. Since this was affected greatly by whatever insulating compound was used for the HV winding, it always was trial and error based on previous designs, modifying the number of layers and therefore number of windings per layer. The head sweep designer at Zenith developed a method mixing a silicone liquid bath with the same dielectric coefficient as the potting compound, so a flyback prototype could be run in a beaker of the stuff instead of actually potting it. This way, if the tuning was off, you could modify the prototype instead of starting from scratch.

You can get by without precise tuning of the HV winding, you just may get more HV variation than a carefully engineered production set.