As far as wire specs go, you can use any "magnet wire" catalog as a reference. I like https://prod.essexwire.com/sites/essexwire.com/files/2017-08/Essex-Wire-Engineering-Data-Handbook-EN.pdf for general wire physical properties, but you will need to check each of the many insulation materials to determine the breakdown voltage for a specific insulation build. Wire is listed in both AWG and metric diameter formats. It's good to have a familiarity with both because you can usually find stock in the AWG sizes.
Sorry to say that after 40+ years I still don't always get my high voltage designs right on the first try. So expect to do some iterating.
First, assuming your core will support the required power, your push-pull primary must be wound carefully to prevent the "flux walking" described in your paper. Assuming this is a step-up transformer, your primary should probably be wound bifilar so that the turns are as symmetrical as they can be, and your drive circuit must provide an exactly 50% duty cycle.
The wire size for the primary should be governed by the required current. You are switching at 250 kHz, but the frequency content of the switched current is higher, so you are limited to pretty fine wire. For round wire, the skin depth at 250 kHz is 0.13 mm, and there is a "rule of thumb" saying to stay below two skin depths, so this is probably a good maximum starting wire diameter. If the resistance is too high and you need more current on your primary, you can consider Litz wire or foil (or just using a heavier gauge of round wire and realizing that you will not get the full effect of the copper in the center of the conductor). Using foil may make it harder to avoid the "flux walking".
Place a layer of Kapton between primary and secondary to keep the high voltage out of your low voltage. Also the winding technique is important. @Neil_UK is correct that a properly wound high voltage transformer will not have high voltage between adjacent windings. However, consider the picture below:
Both are 77-turn windings. However, in example "A" ("layered" winding) turn 41 is adjacent to turn 1. Turn 41 will have more than half (41/77) of the output voltage between the two turns, or more than 3000 volts in your case. So if you use the layered approach, put a layer of Kapton between each layer to provide extra insulation and space. The extra insulation will also reduce interwinding capacitance.
Example "B" is called "bank winding" and is hard to accomplish without a machine. But there are never more than 5 turns between adjacent windings, resulting in reduced inter-turn voltage.
An even worse mistake would be to wind on a toroid and have the high voltage end overlap the low voltage end. When winding a toroid, leave a gap.
You will also need to put extra insulation on the outside of the coil and on the wires leaving the coil.
As a general rule, try to keep the number of turns in both primary and secondary as low as you can to reduce winding capacitance. If you have the space, spreading out the windings is the best approach to prevent interwinding capacitance. Too much capacitance will cause your transformer to ring at its resonant frequency, and your components will have to deal with a spiky, decaying oscillation each time you switch.
Other than these issues, it's pretty much the same as any low voltage switching transformer. Good luck!
