To prove a CLT, I need first to prove that the following approximation holds
$$ \sqrt{n}\sum_{j=1}^n\left(\left|\int_{(j-1)/n}^{j/n}\nu_s\,dW_s\right|\,\left|\int_{j/n}^{(j+1)/n}\nu_s\,dW_s\right|-\nu_{(j-1)/n}^2\,\left|W_{j/n}-W_{(j-1)/n}\right|\,\left|W_{(j+1)/n}-W_{j/n}\right|\,\right)\stackrel{p}{\to} 0,\quad\text{ as }n\to+\infty $$ where $W$ is a Brownian motion independent from $\nu$, which is an adapted continuous bounded stochastic process which has the property $$ \nu_{\Delta} = \nu_{0}+O_p(\sqrt{\Delta}),\quad\forall\Delta>0. $$
I have tried with the expansion (whose correctness is not so clear to me)
and also with the $L^2$-convergence. But, in both cases, I am able to prove only
$$ \sum_{j=1}^n\left(\left|\int_{(j-1)/n}^{j/n}\nu_s\,dW_s\right|\,\left|\int_{j/n}^{(j+1)/n}\nu_s\,dW_s\right|-\nu_{(j-1)/n}^2\,\left|W_{j/n}-W_{(j-1)/n}\right|\,\left|W_{(j+1)/n}-W_{j/n}\right|\,\right)\stackrel{p}{\to} 0,\quad\text{ as }n\to+\infty, $$
i.e without the $\sqrt{n}$ factor.