Odd. There is an article about this in the October M.A.A. Monthly, pages 737-742, by R. Thangadurai and A. Vatwani. They give an elementary argument to show $$ p \leq 2^{\phi(q) + 1} - 1.$$ The best unconditional result they report is T. Xylouris (2009), $$ p \leq c_1 q^{5.2}$$ which improves a 1992 result of Heath-Brown.
Apparently Oesterle showed that GRH implies $$ p \leq 70 q (\log q)^2 $$ which is much better. This was a private communication to the authors, not in the reference list.
Anyway, table of contents at CONTENTS
EDIT: I ran a little computer program for the GRH result, dropping the factor of 70... It certainly appears that the largest prime $q$ for which $ p > q (\log q)^2 $ is $q=5227$ with first prime congruent to $1\pmod q$ being $p=397253 = 1 + 76 \cdot 5227.$ Unprovable. Program run for $q < 8000000$ and print out only $ p > 0.9 \\, q \\, (\log q)^2. $
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./primes_in_progressions
2 3 3.12205
3 7 1.93325
7 29 1.09409
19 191 1.15951
5227 397253 1.03683
170167 24504049 0.992619
4722079 1104966487 0.99082
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$