One of the purposes of "would + bare infinitive" is to express the future in the past, and this is the correct form here.
You should say either
I didn't know that I would have to put my CPR training to use on my own uncle
I didn't know that I would have to put my CPR training to the test on my own uncle
While you could also use the "had to" form here, it would have a different meaning:
You could use the "had to" form when speaking immediately before you perform the CPR (although even here, "would have to" is more likely): "I didn't know that I (had to / would have to) put my CPR training to use on my own uncle - but now I'm about to do so!". "Had to" would express the fact that you initially failed to realise that this was what you were supposed to do, whereas "would have to" would express a sentiment similar to that in your original sentence.
If you failed to apply CPR at all and the police demanded your explanation, you might explain that you somehow didn't realise that this was what you needed to do: "I didn't know that I had to put my CPR training to use on my uncle!".