We need to categorise the types of models before we consider the term $\dfrac{\partial^2 \Sigma}{\partial S \partial \sigma}$. I will only consider local volatility models and stochastic volatility models.
Local volatility models
The local volatility function is, of course,
$\sigma^2(K,T)=2 \dfrac{\partial_T C_{KT}}{\partial_{K}^2 C_{KT}}$
This can be expressed as a function of the implied volatility $\Sigma$,
$$
\sigma^2(K,T) = \dfrac{ 2 \partial_{\tau} \Sigma + \Sigma/\tau }{ K^2 \left[ \partial_K^2 \Sigma - z_1 \cdot \sqrt{\tau} \cdot \left[ \partial_K \Sigma \right]^2 + \left[ 1 / \Sigma \right] \right] \left[ 1/ \left( K \sqrt{\tau} \right) + z_1 \cdot \partial_K \Sigma \right]^2}
$$
where $\tau=T-t$ and $z_1 = \log(S/K)/(\Sigma \sqrt{\tau}) + (1/2) \Sigma \sqrt{\tau}$.
As $\tau \rightarrow 0$, i.e., the short maturity case, we get
$$
\sigma(K) = \dfrac{\Sigma}{1 + \left[ K/ \Sigma \right] \cdot \log(S/K) \cdot \dfrac{d \Sigma}{d K} }
$$
To calculate the term $\dfrac{d \Sigma}{d \sigma}$, notice that $\dfrac{d \Sigma}{d \sigma} = \dfrac{1}{\dfrac{d \sigma}{d \Sigma}}$. Likewise for $\dfrac{d \Sigma}{d S}$ and henceforth for $\dfrac{\partial^2 \Sigma}{\partial S \partial \sigma}$.
You can specify a simple functional form for $\Sigma(K)$, i.e., something like $\Sigma(K):=a \cdot e^{-bK+c}$, hence allowing you to calculate the quantity you are interested in.
Stochastic volatility models
We need to be careful here because there are market models (for the stochastic volatility) and stochastic volatility models. They sound the same but they are not - a stochastic volatility model will not model the implied volatility, but will (obviously) model the stochastic volatility. A market model will model the implied volatility.
Example of stochastic volatility model: SABR model
Example of market model: Schonbucher model
There are papers (Managing Smile Risk paper for SABR model, Schonbucher's 1999 paper for Schonbucher's model) that will be (most likely) out of date - but they will give you good intuition on how to calculate the vanna term for a stochastic volatility model.
The latest papers on perturbation expansions will be proprietary and obviously not public. Both of the papers that I mentioned provide analytic expressions for the implied volatility $\Sigma$ as a function of all relevant parameters. Setting $\tau=0$ or taking the limit $\lim_\limits{\tau \rightarrow 0} \Sigma(\tau,\ldots)$ for the expressions given in those papers (and then calculating the vanna term) should give you the answer you are looking for.
