In this sentence:
テレビをみていた時停電になった - When I was watching TV, the power went out
- how do we know whether to use ながら, or みていた? Do I need to go by the existing Japanese in the sentence and assume its みていた because of the following 時?
Hey! 
ながら means “while”, and 時 means “when/at the time of”
ながら is used for two ongoing concurrently actions, made by the same subject.
In this case, the subject of the first clause( テレビを みていた 時) is not the same as the subject of the second clause(時停電になった).
The first one is the speaker, the second one is the electricity.
Also, the actions aren’t concurrently ongoing, the clause 2 時停電になった is simply an event that happened during the TV watching. - it is not ongoing.
This is one among uses of 時, something like English past simple and past continuous combination “When I was doing something, something2 happened - when I was watching TV, telephone rang”.
ながら typical pattern is “while A, B” “while I was watching tv, I read some newspaper”, “I like to listen to the radio, while sunbathing”.
Also, grammatically, ながら follows verb[stem] rather than verb[past].
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