What is the axiomatic (set theory) context of the P vs NP and NP=EXPTIME conjectures? When the conjecture $\mathbf{P} = \mathbf{NP}$ or $\mathbf{P} \neq \mathbf{NP}$ is set (e.g. by the Clay Mathematical Institute by S. Cook, see here) what mathematical axiomatic system is assumed?

In order to prove or disprove such statements, you need to assume some axioms. Which ones? Only the Peano (2nd order formal language) arithmetic? The Zermelo–Fraenkel set theory with the axiom of choice? Smaller axiomatic set theories (e.g. Gödel's constructible sets, where the continuum hypothesis holds too, see here)?

Obviously, it should be an axiomatic theory that accepts the countable infinite. But which in particular? Is there any published result that would prove them consistent in a particular axiomatic set theory? (In other words, defining a model in which it is true, but not claiming to be true in all models).
It's not specified. When there is a serious enough candidate paper purporting to resolve P ≟ NP, a Special Advisory Committee will be formed to decide whether (and to whom) to award the prize. I presume that the Special Advisory Committee will decide whether your system of axioms is acceptable. If you assume Z-F with choice, I guarantee you they will take it. If you assume P ≠ NP as an axiom, I guarantee you they won't.