Is there an efficient way (an algorithm) to find, given a fixed $n\in\mathbb{N}_+$, all fields of order $p^n$ for all primes $p$ simultaneously?
There's no background behind this question, just an idea.
Given a prime $p$ and $n\in\mathbb{N}_+$ we can find a field of order $p^n$ using an irreducible polynomial $f$ of degree $n$ over $\mathbb{Z}_p[x]$ and considering the quotient $\mathbb{Z}_p[x]/(f)$. Can we achieve something like this but universally for all primes?
By that I mean, if we are able to find certain "thing" in finite time so that that thing will give us a description of all finite fields of order $p^n$ where $n$ is fixed.
I interpret your question as asking whether there is a polynomial $f\in\mathbb{Z}[x]$, of degree $n>1$, such that $\overline{f}$ is irreducible of degree $n$ in $(\mathbb{Z}/p)[x]$, for every prime $p$; unfortunately the answer to this is no. To see this, let $f\in\mathbb{Z}[x]$ be arbitrary. Find some $n\in\mathbb{Z}$ such that $\pm 1\neq f(n)=:m$, and let $p$ be any prime factor of $m$. Then $\overline n$ is a root of $\overline f$ in $(\mathbb{Z}/p)[x]$, since $\overline f(\overline n)=\overline m=\overline 0$, and thus $x-\overline n$ is a factor of $\overline f$, which is hence reducible (or of degree $\leqslant 1$) mod $p$.