It will require adequate impedance in the transformer primaries or the drive will fault as it will be "expecting" to see 10 horsepower motor windings or the equivalent.

This is likely your most difficult challenge.  My guess is that you need inductors between VFD and transformers.  HV transformers are not likely to handle VFD switching-frequency voltage swings, and the VFD is likely to have trouble with HV transformers secondary winding capacitance if not isolated by primary inductors.

I have a box of 30 12KV 1A NTE rectifiers (block mount on heatsink)

At <1kHz, series connection may work.  At 30kHz, the biggest issue I found with diode series strings is variation in reverse recovery charge.  The first diode to recover gets the entire string's reverse voltage, conducting recovery current of other diodes in avalanche breakdown.  Sorting diodes by recovery charge helped, as did using diodes with avalanche energy handling specifications.

BTW, if transformer secondaries are wired as 3-phase, only 6 diodes are required, not 12 as shown in your diagram.

3 strong step-up transformers

If you can find a 3-phase transformer, that will be lighter and more efficient than 3 separate 1-phase transformers.  Not sure where you will find either, nor the HV inductor.  Good luck with that.  (I used resonant charging on my SRSGTC, using intentional inductance of MOTs and a bit more added inductance in primary since MOT inductance was not quite enough.  Charging tuned to 120Hz to match my 60Hz line frequency.  Synchronous, so no diodes.)

The 300 ohm reading was made on the board without 555 and without 0.33uF capacitor because it was the first component that I removed to test its capacity and resistance, and from the measurements it turns out to be good

As a general consideration when chasing intermittent problems, measuring a component to be good is no guarantee.  It might measure good at one time, then fail at another time.

However, since you measured 300 ohms on the board with 0.33uF capacitor removed and 555 removed, the only possibility left is a failure of the socket or board itself.  Some contamination or metal (ie. tin from solder) whisker is intermittently shorting that node to ground.  Are there any spots that were overheated during soldering?  Carbon from charred organic material seems most likely to create 300 ohms.  Metal whiskers are generally lower resistance and ionic conductivity (ie. salty finger prints) are higher resistance.

So the problem is not with the 555 chip. I have made other tests with a multimeter and the only thing that does not fit me is that between pin 2 (and therefore also on 6) and GND I see a resistance on about 300 ohm.

I agree, 300 ohms is likely a problem, and would cause 555 output to stay high.  (Unless your meter measures resistance with unusually high voltage, enough to forward-bias diodes.  To check for this, measure at a higher resistance range such as 20k full-scale.  If it is still 300 ohms, this is likely the cause of your failures.)  I'd guess the failure is internal to 555 chip.  Other less-likely possibility that comes to mind is some resistive short on the circuit board.

Either way, it appears to be intermittent.  Intermittent failures are the hardest to find.  At the time of any given measurement such as repeating the above 300 ohm test, the failure may be present or may not be present.  No easy way around this difficulty.  If you can get the 300 ohm measurement to repeat, measure the chip out-of-socket and measure the socket pins, to determine if the failure is of the chip or board.

However this is the link of the video of the operation, maybe it can be useful to explain me better

Yes, I'd agree with your interpretation.  Looks like interruption is failing to continuously-enabled.  Are you adjusting any potentiometers (50k or 2M) just before operation changes to continuous?  It sounds like the interruption frequency is increasing for about 1 second prior to failing to continuous-enable.  If you were not adjusting frequency (2M pot), then the failure is likely some issue with the 555 circuit.  If you were adjusting frequency, then the failure may be coincidental to the frequency change, so may be unrelated.

If the 50k (pulse width) potentiometer were to fail open (the most common failure), that would result in continuous enable.  Presuming all solder joints are good, adding a jumper from the unused potentiometer pin to the wiper (center) pin avoids the common failure of open wiper.  Then an open wiper failure becomes maximum 50k ohms rather than infinite.  Not too likely to be your specific failure unless you happened to be adjusting both pots simultaneously.

But I have a question: can I test without replacing dead IGBTs?

Yes, but do remove dead IGBTs first.  If the IGBTs failed with gate-source shorted besides source-drain, then leaving them in circuit would overload driver chip.

Do you have access to an oscilloscope?  If so, it shouldn't take too long to figure out the issue.  Probe UCC27425 enable and output pins.  Probe without power to half-bridge, only +5V and +12V supplies on.  Verify that you can adjust enable pulse width and frequency, and that both UCC27425 outputs are low when enable is low.  If that looks good, repeat measurements during operation, set to low duty cycle to minimize risk to IGBTs.

as soon as you turn on the coil, the burts appears "correctly" and after a few seconds it changes without me changing anything

One possibility is a bad connection between socket and IC.  Sockets can be helpful, but also problematic.  Intermittent connection is one possibility for this behavior.

Sure! It starts around 4-5 seconds into it.

Agree with Mads.  I don't see or hear anything indicating a problem.  Might be a transition where the base of the previous arc is hot and ionized enough to make the next enable pulse start immediately with significant arc length.  If bridge voltage and current are still properly phased (ideally bridge output voltage switching slightly before current zero-crossing), it should be fine.

Where the probe is connected (isolated by isolation transformer) :

Some or even most of the ringing may be signal in the FET source lead inductance.  Connect scope ground lead as close to FET body as possible, and route ground lead adjacent probe to minimize loop area (minimize mutual inductance with power wiring).

If I'm seeing images correctly, FET source connection is soldered close to FET body.  That's great.  Best if GDT connection to source doesn't share much if any distance along the wire conducting FET source current.  In other words, solder GDT source wire close to FET, or on FET source lead separately from power (high current) source connection.  Tighter twisting of wires from GDT to FETs and to driver would help as well.  (IMHO, that is more important than moving the resistor to the FET gate lead.)

Reducing half-bridge power ringing will also help.  In case it's helpful, here's my example of low-parasitic-inductance half-bridge construction:
https://highvoltageforum.net/index.php?topic=1324.msg9795#msg9795

By the way, I there any way to predict what would be the perfect primary for my setup.

There are a couple discussions on this forum about optimum DRSSTC primary impedance.  Haven't thought much about SSTC.  Mads (Kaiser Power Electronics) web pages may have guides.

And for the damped wave un my unloaded circuit, It put the fet in activity for quite a long time at the end of each interupter cycle. How can I get rid of it ? I would like to audio modulate it but with that wave, I'm going to have problems !

I think you are talking about the lower-amplitude ring-down after half-bridge drive ends.  That is not likely a problem.  Have you zoomed-in to see the ring frequency?  May be at high frequency, resonance of primary inductance and FET capacitance.

Too much work and winding to risk  smacking it with 60 or 400Hz!

I quite agree.  Given ferrite core and 20kHz design frequency, it is certain to function poorly at 60Hz or 400Hz.  Output voltage capability before saturation will be roughly proportional to frequency.  At 400Hz, max output voltage will be about 100V rather than 5kV.  (I'd misunderstood initially.  Thought you were asking about running a 60Hz transformer at higher frequencies.)

Good idea ! How do you link the disk weight to output voltage ?

Electric field E on top of the sphere (presuming semi-infinite distance to other objects) is V/r, where V is the sphere voltage and r is the sphere radius.  Energy density (J/m^3) of that field is 1/2 * E^2 * e0, where e0 is air (vacuum) permittivity, 8.854pF/m.  Force is energy per unit distance, so J/m^3 is the same as N/m^2.  What matters for voltage measurement is force (weight) per unit area.  As long as the disk is thin and disk diameter somewhat smaller than top-load radius, weight/area is the only needed parameter.  In other words, if you can lift a 1mm thick aluminum disk, it doesn't matter if it is 10mm or 100mm diameter.  Either way, 1mm thick aluminum (2.7g/cc = 2700kg/m^3) is 2.7kg/m^2 for 1mm thickness.

With a bit of algebra, V = r * sqrt(2 / e0) * sqrt(force / area) = r * sqrt(2g / e0) * sqrt(mass / area) where g is gravitational acceleration on earth's surface, 9.8 N/kg.
For MKS units, that becomes:
     V = (1.49MV * r_in_meters) * sqrt(kg / m^2).
For your 0.25m radius top load:
     V = 372kV * sqrt(kg / m^2).
Thus, if you can lift 1mm thick aluminum (2.7kg/m^2), voltage is 372kV * sqrt(2.7) = 611kV.

Measurement weights can be constructed various ways.  Coins are easy if you can find ones with appropriate areal density.  Sharp fields around rims will have some effect.  I've used aluminum foil disks formed to top-load curvature, with various weighting sandwiched between pairs - flattened bits of clay or other putty, coins and double-sticky tape, etc.  Weigh and divide by area to get areal density.

I believe the above is all accurate.  If I've made a mistake, presumably USpring will add a correction or clarification

David, I tried 

Looks like that is working well.

so I just set the 'current' pot to minimum, and checked if there is output voltage.yes there is.

Current limit accuracy (including temperature drift and change with output voltage) is going to be poor down at 2mA.  True for almost any supply capable of 1amp or more.  That power supply circuit powers U1 and other circuitry after current sense resistor R7 (0.47 ohms).  Also, 2mA is only 1mV across R7, well under sensing opamp U3's worst-case input offset voltage specification (+-6mV).

To test leakage current, I'd suggest feeding collector through 10k resistor.  Measure voltage before and after resistor to see how much current the IGBT is drawing.  If more than 40uA, that is a clear violation of a worst-case specified limit, so proof of counterfeit (or damaged) parts.  (Spec. is 40uA at 650V at 25degreesC.)