Averaging Chebotarev's density theorem over families of number fields The Chebotarev density theorem is one of the most celebrated and important results in number theory. We state the following version: for a number field $K$, Galois over $\mathbb{Q}$ with Galois group (isomorphic to) $G$ and a conjugacy class $C$ of $G$, put $\pi(x; C, K)$ to be the number of prime ideals $\mathfrak{p}$ in $\mathcal{O}_K$ which is unramified in $K$ and for which the Frobenius $\sigma_{\mathfrak{p}} = C$. Then $\pi(x; C, K)$ satisfies 
$$\displaystyle \pi(x; C, K) = \frac{|C|}{|G|} \text{Li}(x)(1 + o(1)).$$
The original proof given by Chebotarev is ineffective, so that the determination of the error term $o(1)$ above is not tenable. Various effective improvements have been proved over the past several decades. 
Conditioned on GRH, one can obtain the expression
$$\displaystyle \pi(x; C, K) = \frac{|C|}{|G|} \text{Li}(x) + O \left(x^{1/2} [K : \mathbb{Q}] \log(|\Delta_K x|)\right),$$
where $\Delta_K$ is the discriminant of $K$. Thus, assuming GRH and running over a family $\mathcal{F}$ of fields $K$ of equal degree and the same Galois group with the property that for any integer $n$ there is a uniformly bounded number of fields $K$ in $\mathcal{F}$ with discriminant $n$, one has
(1)
$$\displaystyle \sum_{\substack{n \leq Q \\ |\Delta_K| = n \\ K \in \mathcal{F}}} \sum_C \left \lvert \pi(x; C, K) - \frac{|C|}{|G|} \text{Li}(x) \right \rvert = O \left(Q x^{1/2}  \log(xQ) \right).$$ 
Thus, the error term remains acceptable (i.e., $o(x)$) even when $Q$ is as large as $x^{1/2} (\log x)^{-A}$, for $A > 1$. 
The latter statement is similar to a famous result of Bombieri and Vinogradov. For a positive integer $q$ and an integer $a$ such that $\gcd(a,q) = 1$, put $\pi(x; a, q)$ for the number of (rational) primes $p \leq x$ such that $p \equiv a \pmod{q}$. Then Bombieri-Vinogradov theorem asserts that
$$\displaystyle \sum_{q \leq Q} \max_{y < x} \max_{\substack{1 \leq a \leq q \\ \gcd(a,q) = 1}} \left \lvert \pi(y; a, q) - \frac{1}{\phi(q)} \text{Li}(y) \right \rvert = O \left(Q x^{1/2} (\log x)^5\right),$$
so we may take $Q$ to be as large as $x^{1/2} (\log x)^{-A}$ for $A > 5$. 
The Elliott-Halberstam conjecture is the statement that one can take $Q$ as large as $x^{1 - \varepsilon}$ for any $\varepsilon > 0$ in the statement of Bombieri-Vinogradov. 
Can one prove (1) without assuming GRH? Further, can one expect that the 'level of distribution' in (1) can in fact be taken as large as in the Elliott-Halberstam conjecture?
The most basic example of where this question may apply is the family of (real or imaginary) quadratic fields. 
 A: Let us consider the related problem of finding a suitable $\delta>0$ such that
$\displaystyle\sum_{\substack{q\leq x^{\delta-\epsilon} \\ K\cap \mathbb{Q}(e^{2\pi i/q}) = \mathbb{Q}}}\max_{(a,q)=1}\Big|\sum_{\substack{p\leq x \\ p\equiv a\pmod{q} \\ [\frac{K/\mathbb{Q}}{p}]=C}}1 - \frac{|C|}{|G|}\frac{\mathrm{Li}(x)}{\varphi(q)}\Big|\ll \frac{x}{(\log x)^A}$.
Let $H$ be an abelian subgroup of $G$ such that $H\cap C$ is nonempty, and let $E$ be the fixed field of $H$.  Ram Murty and Kumar Murty proved that this holds when  $\delta = \frac{1}{\max\{2,[E:\mathbb{Q}]-2\}}$.  The strong Artin conjecture for the irreducible representations $\rho$ of $G$ would imply that one can replace $[E:\mathbb{Q}]$ with $\max \rho(1)$.
Here, the averaging is much simpler than you propose in your initial problem.  We are simply averaging over Dirichlet characters and have an optimal version of the large sieve.  We have a level of distribution $\delta = 1/2$ when $[E:\mathbb{Q}]\leq 4$, which includes the cases where $K/\mathbb{Q}$ is abelian or dihedral.
If $K/\mathbb{Q}$ is "sufficiently nonabelian", then the level of distribution is quite small.  Ultimately, this is related to the lack of strong bounds for Artin $L$-functions in the critical strip; we're using the convexity bound and Phragmen-Lindelof.  So the level of distribution will be proportional to the reciprocal of the largest degree of all of the $L$-functions (when viewed over $\mathbb{Q}$) under consideration.  I find it hard to believe that a proper analogue of Bombieri-Vinogradov (with a level of distribution equal to $1/2$) is possible without some serious advance toward Lindelof for Artin $L$-functions twisted by Dirichlet characters.  If one cannot establish a proper BV in this setting, I don't see how one could do it in the more complicated setting you propose.  This also ignores many subtleties in averaging over number fields with a fixed nonabelian Galois group, many of which are catalogued very nicely in the work of Pierce, Turnage-Butterbaugh, and Wood arxiv.org/abs/1709.09637 (as Alison already mentioned).
Also, a small point:  You don't necessarily have that $|\{K\in \mathcal{F}\colon |\Delta_K|\leq Q\}|\ll Q$.  So the RHS of your equation (1) might be off, depending on what $\mathcal{F}$ is.
