It's a blocking oscillator, Q454 is actually going to control it (frequency modulation, sort of; more on that later). The feedback winding will be sized to give ca. 1-5V. Less and there's insufficient gain for hard switching, more and V_EB is exceeded, drawing bias current through that path, and Q458 literally beats itself to death (or until R458 fuses).
Primary peak voltage is normally about double the supply voltage, but you might design it for more just so that you can save some on secondary turns, since the voltage is high. You can't go too far, because again, the feedback needs to be okay, and a large ratio of peak voltages (turn-on vs. turn-off) gives less feedback voltage while still avoiding E-B breakdown when off.
This is basically how things go, but the high voltage winding very likely has quite a lot of capacitance, and so rather than a nice square flyback pulse, you get a slow wumpy resonant (or quasi-resonant) waveform instead.
In a well damped blocking oscillator (and I will note R457-C457 will give some good damping, if the primary inductance is about what I would guess it to be), the transistor turns on, then turns off, and just stays off, even after the output rings down. Because if it's well-damped, it doesn't ring down at all, it just kinda...and it's done. Back to zero. But if it's underdamped, after the flyback pulse the voltage keeps swinging, and about when it completes that first cycle, it can kick the transistor back into conduction. And so on. It runs for a burst of pulses, rather than just one. This is called squegging.
How long the transistor runs for, depends on C455. The feedback winding pushes on the transistor base, and pushes against C455, discharging it. If it's fully discharged (i.e. by several volts) during a single pulse, it won't squeg; if Q454 is supplying enough current to keep its voltage up, it will run CW (continuous).
So, this may be a resonant blocking oscillator, in which case Q454's control kinda looks more like a variable on-time control, but frequency is changed a bit too, as it looks kinda quasi-resonant (class E). It still rings down and does some kind of pulse-skipping or burst mode operation at really low currents (possibly at really low beam intensity??).
The resonant frequency is controlled by the capacitance of the secondaries (and the rectifier diodes), and the transformer's core gap (and thus winding inductance). You'll need a gapped ferrite core here, but the gap probably won't need to be very much, since you'll want to use a relatively large core to keep the V/turn up, so you don't end up needing ten thousand goddamn turns or whatever.
Good luck winding the secondary -- it only needs to supply ~1mA so the wire can be very fine indeed. If you don't have, or aren't comfortable winding, wire this fine (>40AWG?), you may find you need a far larger transformer than the original. And, if this is a repair thing, you really ought to just find an original part; but if this is for just demonstrating the HV supply for its own sake (and presumably for some purpose, perhaps making a CRT glow?) it's not bad, but keep this in mind. (Upside, I guess: you could use a much bigger transistor, and beefier primary, and kick some serious current out of the thing.)
TL;DR well, if you know the primary, assume peak about equal to supply and wind secondaries accordingly. Mind core size and gap. Hand-waving notes on operation and design of a blocking oscillator, not much for numbers or equations but more like hints at how to figure out what equations would apply.
Tim
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