[Arran]

Did you check the power cord for continuity? One end should go to the power switch and the other to one end of the transformer primary. How about the power switch itself? If you read nothing between the plug and the power switch then the power cord is likely shot, at least at the plug end. If you read something then check across the switch as mentioned before. Another thing you could try is to bypass the power switch by shorting across it with an allegator clip test lead, or by soldering a length of wire across it.
Generally speaking if the transformer is bad it will smell and appear charred along with tar dripping out of it. All of the other windings check out except the primary so either the primary winding is open or something between it and the power line is open, since power transformers rarely fail on the primary side with anything short of a lightning strike and it is not uncommon for an old radio to have a rotted power cord or one of these Philco models to have a bad power switch I would bet against the transformer being the problem.
Regards
Arran