Is determining if there is a prime in an interval known to be in P or NP-complete?
So your problem is as follows:
Input: integers $\ell,u$
Question: does there exist a prime in $[\ell,u]$?
As far as I know, it is not known whether that problem is in P or not.
Here's what I do know:

Primality testing (given a single number, test whether it is prime) is in P, so if the range is small enough, you can exhaustively test each number in the range to see whether it is prime -- but this does not lead to a general algorithm.

If Cramer's conjecture is true, then the problem is in P.  Cramer's conjecture says that the gap between consecutive primes near $n$ is $O((\log n)^2)$, so the following algorithm will be in P: iterate through the numbers $\ell,\ell+1,\ell+2,\ell+3,\dots$, testing each whether it is prime; if you find one that is prime, stop immediately with a yes answer; if you hit $u$, stop with a no answer.  Cramer's conjecture tells us that you'll stop after most $O((\log \ell)^2)$ primality-tests, so that algorithm will run in polynomial time.
Unfortunately, the known results on prime gaps don't seem strong enough to prove unconditionally that the problem is in P.

Here is another simple algorithm: repeatedly pick a random integer $r$ from $[\ell,u]$, and test whether it is prime.  Stop if you find a prime, or if you've tried every integer in $[\ell,u]$ and none were prime.  Heuristically, we should expect this to be efficient in practice.  The prime number theorem tells us that if you randomly choose number in the vicinity of $u$, the probability it is prime will be about $1/\log n$.  Thus, heuristically, we should expect that after about $O(\log u)$ iterations, you'll typically find a prime and stop, if one exists.  On the other hand, due to the coupon collector's problem, if there is no prime in the range $[\ell,u]$, you'll stop after about $O((u-l) \log(u-l))$ iterations.  So, if we had a good upper bound on the size of the longest gap between prime numbers, this would imply that the problem is in BPP.  Even without such an upper bound, it follows that random problem instances are easy.

Probably one can apply sieving methods to improve the running time in practice (e.g., to avoid doing any primality testing on numbers that are divisible by a small prime).  I don't know whether this can be shown to lead to any asymptotic improvement.

Due to these techniques, the problem is probably easy in practice.

Due to the above remarks, I personally doubt that the problem is NP-complete.