Could be the transformer, could be something else..
I checked the transformer and it has no ac voltage from the secondary but does have continuity across the secondary side.
how are you measuring continuity? By resistance or by the beep! Audio sound.
-by the beep! Sound, you will only get a beep on the secondary since it has less resistance than a 240V primary.
-if by ohms, in the primary should be low resistance. while on the 240 you should get high ohms.
It's a possibility that it could be a diode... or a capacitor... capacitors are notorious to fail and then diodes... imo.
Make sure the transformer works first... if it is ok, you can post a picture here of the electronics and I might help you test our components down the line... it takes time though... good luck.
As for the transformer... I would guess any transformer that is around the same size with same amp rating (although you don't know the amp rating it should not be that much... i would guess a 20amp.. or less) and primary plus output rating should work.