On the TV chassis power switch: One set of contacts should be normally open and the other normally closed. So you need to verify that you have two working sets of contacts(one NO and one NC), and that they both alternate between open and closed when you actuate the TV power switch on the TV volume control. Next make sure the set of contacts that's closed (with the TV switch turned on) connects one side of the AC line voltage to the TV power transformer primary. I'm guessing someone may have unsoldered your switch and has those two sets of contacts exchanged.
But here's the confusing part: There's also a similar dual set of contacts on the radio tone/on-off switch. So when the radio is turned on, one set of the radio contacts also turns on AC line voltage to the primary of the TV chassis power transformer. So you should also check that that set of contacts is not stuck in the closed position (when the radio is turned off). It's also possible to have both the radio and TV turned on at the same time, in which case the TV power transformer gets AC line voltage, but both the radio and TV have selected tube filaments turned off, and so neither should work(I think). Clear as mud?