Whats wrong with this model theoretic proof of the twin primes conjecture?

It is not correct to conclude that $\Psi$ is inconsistent. It is true that there is no assignment of values to the $c_i$ in the standard model $\mathbb{N}$ which makes it a model of $\Psi$. However, that doesn't mean that $\Psi$ is inconsistent, because there could be some different model of it (with a nonstandard underlying model of natural numbers). Indeed, your argument proves $\Psi$ is finitely satisfiable, so by compactness such a model exists.

Here is a simpler example of the same phenomenon. Add just a single constant $c$ to the language of arithmetic and consider sentences $\varphi_n$ saying that $c>n$ for each $n\in\mathbb{N}$. Then $T\cup\{\varphi_n:n\in\mathbb{N}\}$ is finitely satisfiable, but again there is no way to give $c$ a value in $\mathbb{N}$ to make it a model. Instead, to get a model you need to take a nonstandard model of $T$ which has elements which are greater than every standard natural number.

In fact, not only is your theory consistent, but (regardless of the status of the twin primes conjecture) in any nonstandard model of arithmetic we can find interpretations of the $c_i$ (we can even do that with countably many $c_i$, not just $n+1$).

The point is that the factorial function makes sense in any nonstandard model of arithmetic, using Gödel's argument to code recursive functions within the language of arithmetic, see here. Let $M$ be a nonstandard model of arithmetic and let $I$ be an infinite (i.e., nonstandard) number in $M$. For $i\in N$, let $c_i=(I+i)!-1$. The $c_i$ so defined are distinct, and the usual (Euclid's) argument for the infinitude of primes shows that no standard prime divides any of the $c_i$ or of the $c_i+2$. Note that absolutely nothing changes if we have countably many $c_i$ rather than only $n+1$.

(Now, if the twin primes conjecture is false, and $N$ as in your post, then in no model of your theory is any of the $c_i$ a prime number. But this is not an issue.)