How do you call the output terminals of that transformer then? Are they both live?
If it's just a plain transformer with no case or receptacle, then the output is floating with respect to ground and neither conductor is "hot" or "neutral". They're just two wires with 120V between them. If the transformer is in a case with a receptacle and cord, then one of the output terminals will probably be bonded to ground. That's the neutral. The one with 120V with respect to ground is hot.
All your 12V DC adapter needs is 120V between the two prongs.
It would still be easier to just get a 240V to 12V adapter and put DC the connector from the old adapter on it so it fits your device.