Taylor expansion of stochastic variables with dynamics of the form $dX_t=b(\sigma_t,X_t)dW_t$

https://www.math.nyu.edu/~cai/Courses/Derivatives/compfin_lecture_5.pdf

In the above document stochastic taylor expansions are nicely explained.

Let us now consider a typical SDE model in finance like SABR. here the form is:

$$dS_t=b(\sigma_t,S_t)dW_t$$

My point is that because $$b$$ also depends on another stochastic variable, we cannot follow the simple steps provided in the document. Or can we ....... So my question, how dos the Taylor expansion of $$S_t$$ look like in this setup? To keep it simple we can just say first order, that is not not important.

And also: can we even derive the Taylor expansion if we dont know the dynamics of $$\sigma_t$$. I wonder if we can just say $$dS_t=\sigma_t b(S_t)$$ and treat $$\sigma_t$$ as a "constant". How wrong would that be?