I deal with transformers a lot.. some are better than others.
Mike is correct..MOST transformers will fail on the primary side due to too much draw on the 2ndary. Transformers i deal with have an output of between 6KV and 15KV at 30 or sometimes 60mA. When the primary fails, the transformer cooks rather like what you have there.. winding laminations break down due to heat, transformer shorts and its goodbye primary winding. Ive seen it on the 2ndary also.. this is ugly as the high voltage goes bang and blows a bloody great hole in the transformer lid. Primary failure is very common if its calibrated right to spec or too close.. we always calibrate loading for modern transformers to 80% give or take 5% of open circuit current... go above this and you run them hard... go below and you cause excessive heat... the need for smaller and smaller profiles on these units has seen a decrease in reliability.

All transformers work on the same principals.. i have some transformers on the market that if they run open circuit will burn out pretty quickly... i have others that will happily run open circuit for a couple of years... its all due to the quality of manufacture... some will overheat for no real apparent reason after approx 8 - 10 years of use.. others will last a lifetime.. all down to the quality of manufacture. I know of one Chinese brand that will simply burn out after approx 2 years.. you can almost time your clock to them.
Check your bridges as suggested, but, it could also be that the transformer died ( technical term in my field when explaining to a customer).. it does happen.. not often in a pinball transformer i admit but it is possible.

I thought it may have been the case when the first transformer was smoking, but after putting the second transformer in and it started smoking instantly, I knew it was something else other than the age of the transformer.

Now this has happened, do you think the transformer is still usable in my South Park machine or if it has started smoking, does that mean it's time to get a new transformer for both machines (when I have X-Files sorted out)?

I am currently trying Mike's fault-finding tips and also looking into the fluoro tube.

Thanks,
Adam.