By the results of

*Baker, R.C.; Harman, G.; Pintz, J.*, **The difference between consecutive primes. II**, Proc. Lond. Math. Soc., III. Ser. 83, No.3, 532-562 (2001). ZBL1016.11037.

the number of primes in the interval $[x, x+x^{0.525}]$ is $\gg x^{0.525}/\log x$ for $x$ large enough (see the final inequality of the paper); in particular, $p_{(n-1)^2} + p_{(n-1)^2}^{0.525} \geq p_{n^2}$ for $n$ large enough. This gives a bound of the form $O( n^{1.05} \log^{0.525} n )$. Any improvement on this bound would likely improve the Baker-Harman-Pintz bound on large prime gaps, so I doubt one can do much better using what is in the literature. Of course, the situation is much better on RH or even on LH (the Lindelöf Hypothesis). (Also, one can show the expected bound of $O(n \log n)$ for almost all $n$, with a fairly small exceptional set, using the known results on primes in short intervals on average.)