Just about every particle physics computation is done via analytic continuation from Euclidean signature, either via Euclidean lattice simulations or via the $+i\epsilon$ prescription in perturbation theory. So in that sense, yes, Wick rotation is always valid for QFTs. (There are a handful of cases like Chern-Simons theory where the analytic continuation gets rather subtle.)
This observation -- that QFT computations are really Euclidean computations -- predates the Osterwalder-Schrader Theorem. The Osterwalder-Schrader theorem is one attempt for formalize it within a particular setting. The theorem is a tool; it tells you that analytic continuation works if your Euclidean model satisfies the axioms. But it's not necessary for a theory to satisfy the OS axioms in order to exhibit a link between its Euclidean and Lorentzian versions.
That said, most of the OS axioms are pretty uncontroversial: Analyticity, Euclidean invariance, reflection positivity, and ergodicity are very reasonable demands. Any Lorentzian QFT we use in the real world should satisfy them, at least in spirit. (QED, for example, has a Landau pole, and probably doesn't exist non-perturbatively, but one can still work with it as a formal perturbation series.)
The other axiom -- regularity -- is "technical", meaning that it's not really clear if it's physically necessary or just a convenient mathematical tool. I wouldn't want to bet my life on QCD satisfying an appropriate analogue.