In this case, the essential boundary condition is Neumann and natural is Dirichlet. Since Neumann is essential and is nonhomogeneous, you can use the concept of lift, referred by a couple of authors, for example Demkowicz book about HP finite elements.
Follow explanation from Demokowicz book.The $\tilde {\mathbf{q}}(x)$ is a lift on Neumann data $g$. Lift $\tilde {\mathbf{q}}(x)$ coincide with the data on boundary if $x \in \partial \Omega _N$. In other words, lift is an extension of $g$ to the whole domain.
With lift at hand, we can subtract it from the solution, and the difference will vanish on the Neumann boundary. The collection of such sums is identified as the algebraic sum of lift $\tilde {\mathbf{q}}(x)$ and space $H^\textrm{div}_0(\Omega)=\{\mathbf{q} \in H^\textrm{div} \,|\, \eta \mathbf{q}(x)=0\;x\in\partial\Omega_N \}$, and called an affine space:
\begin{equation}
\tilde {\mathbf{q}} + H^\textrm{div}_0(\Omega) = \left\{
\tilde {\mathbf{q}} + \mathbf{q}: \mathbf{q} \in H^\textrm{div}_0(\Omega)
\right\}
\end{equation}
The final variational formulation can be expressed in variational formulation
\begin{equation}
\left\{
\begin{array}{l}
\mathbf{q} \in \tilde {\mathbf{q}} + H^\textrm{div}_0(\Omega) \\
u \in L^2(\Omega) \\
(\mathbf{q},\tau) + (u,\nabla \cdot \boldsymbol\tau) = 0\quad\forall \boldsymbol\tau \in H^\textrm{div}_0(\Omega) \\
(\nabla \cdot \mathbf{q}, v) = 0\quad\forall v \in L^2(\Omega)
\end{array}
\right.
\end{equation}
That can be formulated in another way, more convenient for implementation. Once we have found particular function $\tilde{\mathbf{q}}(x)$ that satisfies the nonhomogeneous Neumann data, we can simply make substitution $\mathbf{q} = \tilde{\mathbf{q}} + \mathbf{w}$ where $\mathbf{w} \in H^\textrm{div}_0(\Omega) $ satisfies homogenous boundary data, that corresponds to formulation
\begin{equation}
\left\{
\begin{array}{l}
\mathbf{w} \in H^\textrm{div}_0(\Omega) \\
u \in L^2(\Omega) \\
(\mathbf{w},\boldsymbol\tau) + (u,\nabla \cdot \boldsymbol\tau) = -(\tilde{\mathbf{q}},\boldsymbol\tau) \quad\forall \boldsymbol\tau \in H^\textrm{div}_0(\Omega) \\
(\nabla \cdot \mathbf{w}, v) = -(\nabla \cdot \tilde{\mathbf{q}}, v) \quad\forall v \in L^2(\Omega)
\end{array}
\right.
\end{equation}
At that point, one can see that there is no difference what we do in classical elements and mix formulation, only we need to identify what is essential boundary condition, which has to be satisfied a priori. Practically we eliminate DOFs, and calculating residuals.
However, is small technical difference, in classical finite element formulation DOFs have physical meaning, and we used to express $\tilde{\mathbf{q}}$ in the space of finite elements by setting nodal values.
In this case like here, for convenience one like to approximate $\tilde{\mathbf{q}}$ on finite element base functions. That simplify implementation. It can be noticed that for RT elements, attention can be focussed on faces on Neumann boundary. For each face where Neumann boundary is applied, small face-by-face problem can be solved to find $\tilde{\mathbf{q}}^h$ as a linear combination of finite element base functions
\begin{equation}
<\eta\cdot\gamma(\boldsymbol\tau^{h,f}),\eta\cdot\gamma (\tilde{\mathbf{q}}^{h,f})-g>_{\partial\Omega^{h,f}_N} = 0 \quad\forall \gamma(\boldsymbol\tau)^{h,f}
\end{equation}
where $f$ is face index where boundary condition is applied and $\gamma$ is a trace operator, and trace of base function on face is piecewise polynomial for RT element.
EDIT:
I have not found implementation or paper for a general transient nonlinear problem with nonhomogenous Neumann bc like this, but we manage to implement this with hierarchical spaces in MoFEM (I am the one developers of that code), which can handle such problems easily. So you can look how is done.
Practical implementation for finding lift is here,
http://mofem.eng.gla.ac.uk/mofem/html/struct_mix_transport_1_1_mix_transport_element_1_1_op_evaluate_bc_on_fluxes.html#ad175a300e635a0f472a947373a2e7b43
This is applied to time dependent nonlinear problem of fluid flow in unsaturated soil
http://mofem.eng.gla.ac.uk/mofem/html/mix_us_flow.html
