(How) Could we discover/analyze NP problems in the absence of the Turing model of computation?
You may wish to look at cost semantics for functional languages. These are various computational complexity measures for functional languages that do not pass through any kind of Turing machine, RAM machine, etc. A good place to start looking is this Lambda the Ultimate post, which has some good further references.

Section 7.4 of Bob Harper's Practical Foundations for Programming Languages explains the costs semantics.

The paper On the relative usefulness of fireballs by Accattoli and Coen shows that $\lambda$-calculus has at most linear blowup with respect to RAM machine model.

In summary, on this other planet things would be pretty much the same with regards to NP, but there would be fewer buffer overflows, and there wouldn't be as much garbage lying around.