A possible answer is that in fact the wiki article already sums it up pretty well: https://en.wikipedia.org/wiki/Propagation_of_uncertainty.
In general you need the partial derivatives, or in the multidimensional case the Jacobian (matrix). If you know the function, this is rather straightforward an exercise in calculus. In case of a linear function, the derivatives become trivial.
In the general case of an arbitrary function, you can also chose from several other options. The article lists five variants; my practical knowledge extends to two of these. The accuracy and feasibility depend on the type of function and the kind of approximation you need.
1) Series evolution: you can do the Taylor expansion and stop at your choice of order. This introduces a shift in the expectation value the propagated error w.r.t the true error (of an infinite series) -- it becomes biased. Given a well behaved function, this might be negligible.
2) Monte-Carlo simulation. Has the benefit that you might not even have to know the actual analytical (if existing) form of your function. But this becomes an own branch of research, depending on the function. Keywords here are efficiency and accuracy of the calculation.
The other methods are more specialized, mostly to problems that can't be well approximated by the two above.
So, concerning your original question: to disentangle the uncertainty $\delta y$, simply write down the full Taylor expansion term. For the first order (which is sufficient in most cases) have a look at the wiki page.