Let $K$ be a number field, and $p$ be a rational prime. Then the Chebotarev Density Theorem implies we can find primes $v$ and $w$ of $K$ of degree 1 which are split and nonsplit respectively in $K[\sqrt{p}]$. What is the best known effective (upper) bound for the norms of the least such primes (not assuming GRH)? In particular, is there a bound which is asymptotically strictly less than $\sqrt{p}$ (times a constant coming from the field $K$)?

EDIT: I'd like to clarify, in response to the comments below. The situation I'm wondering about is when we fix K, and let p vary. So when K is a cyclotomic field (adjoin, say, the qth root of unity for a prime q), I'm asking about the least prime which is a quadratic nonresidue (resp. residue) mod p, which is 1 mod some fixed prime q, and I'm hoping that there is a bound of the form $\sqrt{p}$ times (something in terms of q). Under GRH this is true --- in fact under GRH, we can get a bound of the shape $(\log p)^2$ times constants coming from K.