[Arran]

Before spending a pile of money or going on a massive parts search I would make absolutely sure that the transformer is broiled, disconnect everything from the secondary side and hook up a light bulb in series with the primary. If the transfomer is find the lamp will light up briefly when you apply power then drop down to a dim level, if one of the secondary windings is shorted internally the lamp will not dim much if at all. I would also make a few ohm meter checks, the high voltage winding should read near equal DC resistance on both sides of the center tap, one side should be within 20-25 ohms of the other, if one side is substantially lower then the winding is shorted. I would also check the wiring in the rest of the set between B+ and B- which the ohm meter in case there is an external short.
I hate dealing with bad transformers, fortunately most of the bad ones I have found are in sets from the 1940s when things were standardized a little more, ones with 300-0-300, 5 volt, and 6.3 volt windings where they are easier to find.
Regards
Arran