At the request of Andrej and PhD, I am turning my comment into an answer, with apologies for self-advertising.
I recently wrote a paper in which I look at how to prove the Cook-Levin theorem ($\mathsf{NP}$-completeness of SAT) using a functional language (a variant of the λ-calculus) instead of Turing machines. A summary:
- the key notion is that of affine approximations, i.e., approximating arbitrary programs by affine ones (which may use their inputs at most once); the intuition is that
$$\frac{\text{Boolean circuits}}{\text{Turing machines}}=\frac{\text{affine $\lambda$-terms}}{\text{$\lambda$-terms}}$$
so affine $\lambda$-terms approximate arbitrary computations arbitrary well just like Boolean circuits;
- the upshot is that, in the $\lambda$-calculus world, the proof is much "higher-level", it uses order theory, Scott-continuity, etc. instead of hacking Boolean circuits; in particular, the slogan "computation is local" (which is given by many as the message underlying the Cook-Levin theorem) becomes "computation is continuous", as expected;
- however, the "natural" $\mathsf{NP}$-complete problem is not CIRCUIT SAT but HO CIRCUIT SAT, a kind of "solvability" of linear λ-terms or, more precisely, linear logic proof nets (which are like higher-order Boolean circuits);
- of course, one may then reduce HO CIRCUIT SAT to CIRCUIT SAT, thus proving the usual Cook-Levin theorem, and the gory, low-level details are all moved to building such a reduction.
So the only thing that would change "on this side of the planet" is, perhaps, that we would have considered a λ-calculus-related problem, instead of Boolean-circuit-related problem, to be the "primordial" $\mathsf{NP}$-complete problem.
A side note: the above-mentioned proof could be reformulated in a variant of Accattoli's $\lambda$-calculus with linear explicit substitutions mentioned in Andrej's answer, which is perhaps more standard than the $\lambda$-calculus I use in my paper.
Later edit: my answer was just a bit more than a cut-and-paste from my comment and I realize that something more should be said concerning the heart of the question which, as I understand it, is: would it be possible to develop the theory of $\mathsf{NP}$-completeness without Turing machines?
I agree with Kaveh's comment: the answer is yes, but perhaps only restrospectively. That is, when it comes to complexity (counting time and space), Turing machines are unbeatable in simplicity, the cost model is self-evident for time and almost self-evident for space. In the $\lambda$-calculus, things are far less evident: time cost models as those mentioned by Andrej and given in Harper's book are from the mid-90s, space cost models are still almost non-existing (I am aware of essentially one work published in 2008).
So, of course we can do everything using the purely functional perspective, but it is hard to imagine an alternative universe in which Hartmanis and Stearns define complexity classes using $\lambda$-terms and, 30 to 50 years later, people start adapting their work to Turing machines.
Then, as Kaveh points out, there is the "social" aspect: people were convinced that $\mathsf{NP}$-completeness is important because Cook proved that a problem considered to be central in a widely studied field (theorem proving) is $\mathsf{NP}$-complete (or, in more modern terminology, using Karp reductions, $\mathsf{coNP}$-complete). Although the above shows that this may be done in the $\lambda$-calculus, maybe it would not be the most immediate thing to do (I have my reserves on this point but let's not make this post too long).
Last but not least, it is worth observing that, even in my work mentioned above, when I show that HO CIRCUIT SAT may be reduced to CIRCUIT SAT, I do not explicitly show a $\lambda$-term computing the reduction and prove that it always normalizes in a polynomial number of leftmost reduction steps; I just show that there is an algorithm which, intuitively, may be implemented in polynomial time, just like any complexity theorist would not explicitly build a Turing machine and prove it polytime (which, let me say it, would be even crazier than writing down a $\lambda$-term, let alone check for mistakes).
This phenomenon is pervasive in logic and computability theory, complexity theory just inherits it: formal systems and models of computation are often used only to know that one can formalize intuitions; after that, intuition alone is almost always enough (as long as used carefully). So reasoning about the difficulty of solving problems like SAT, TAUT, SUBSET SUM, GI etc., and thus developing the theory of $\mathsf{NP}$-completeness, may largely be done independently of the underlying computational model, as long as a reasonable cost model is found. The $\lambda$-calculus, Turing machines or Brainfuck programs, it doesn't really matter if you know that your intuitions are sound. Turing machines gave an immediate, workable answer, and people didn't (and still do not) feel the need to go further.
