$\newcommand{\tnu}{\tilde\nu}$\newcommand{\tnu}{\tilde\nu}
Continuing Alex Ravsky's comment, we have 
\begin{equation*}
	2^{2^n}\tnu_{n,1}=\sum_{j=0}^{2^n-1}S_j=S_0+(2^n-1)S_1, \tag{1}
\end{equation*}
where 
\begin{equation*}
	S_j:=\sum_{F\subseteq[2^n]}\Big|\sum_{i\in F}a_{i,j}\Big|,
\end{equation*}
and for each $j\ne0$
\begin{equation*}
	S_j=S_1=\sum_{k,l=0}^M\binom Mk \binom Ml |k-l|,
\end{equation*}
where 
\begin{equation*}
	M=M_n:=2^{n-1}. \tag{2}
\end{equation*}

So, 
\begin{equation*}
	S_1=2^{2M} E|K-K'|, \tag{3}
\end{equation*}
where, for each natural $n$, $K=K_n$ and $K'=K'_n$ are independent random variables (r.v.'s) each with the binomial distribution with parameters $M,1/2$. By the central limit theorem, the distribution of $(K-M/2)/\sqrt{M/4}$ converges weakly to the standard normal distribution (as $n\to\infty$), and hence $V_n:=(K-K')/\sqrt{M/4}$ converges in distribution to $Z-Z'$, where $Z$ and $Z'$ are independent standard normal r.v.'s. 

Also, $EV_n^2=2<\infty$ for all $n$ and hence, by the [de la Vallée-Poussin theorem][1], the $V_n$'s are uniformly integrable. 

Therefore (see e.g. [Theorem 3.5, p. 31][2]), $E|V_n|\to E|Z-Z'|$ and hence, by (3), 
\begin{equation*}
	S_1\sim 2^{2M}  \sqrt{\frac M4}\;E|Z-Z'|=2^{2M}  \sqrt{\frac M4}\;\frac2{\sqrt\pi}. \tag{4}
\end{equation*}

Also,
\begin{equation*}
	S_0=\sum_{F\subseteq[2^n]}\sum_{i\in F}1=\sum_{k=0}^{2M}\binom{2M}k k=2^{2M}M. \tag{5}
\end{equation*}
Collecting the pieces (1), (5), (4), and (2), we finally get 
\begin{equation*}
	\tnu_{n,1}\sim\frac1{\sqrt{2\pi}}\,2^{3n/2}.
\end{equation*}

[1]: https://en.wikipedia.org/wiki/Uniform_integrability#Relevant_theorems 
[2]: http://cermics.enpc.fr/~monneau/Billingsley-2eme-edition.pdf