Connect a square wave generator to the primary (or the cal output from a CRO). if you view the waveform, it should be a series of decreasing sine waves at a high frequency. if it has only one major narrow pulse it means that a winding has a short circuit on it.
If you have a variable frequency source, setting the frequency to the operating frequency of the transformer, it should act like a transformer i.e. each winding should have a voltage on it proportional to the number of turns.
Very high voltage windings are made with very thin wire and are extremely unreliable, the line output transformers in old fashioned CRT type televisions were responsible for about 10% of all their faults!!
Frank