Actually this is not very difficult. Maybe the term "Jacobian of the svd" is a bit misleading since the svd is a function that "outputs" three matrices from one matrix input. Hence the derivative of such a construct would be three four-dimensional objects. So calling this derivative a "Jacobian" is strictly speaking wrong, because the derivative is not a (two dimensional) matrix. Nevertheless, this is what the paper is about.
It can be done because the outputs can be regarded as functions of the elements $a_{nm}$ of the input matrix $\mathbf A$, e.g. for the $i,j$-th element of the $\mathbf U$ matrix you have $u_{ij} = u_{ij}(a_{11},a_{12},a_{13},\ldots,a_{NM})$
Hence you can also obtain all partial derivatives (if they exist!) of the output matrix elements from input elements: $\frac{\partial u_{ij}}{\partial a_{nm}},\frac{\partial s_{ij}}{\partial a_{nm}},\frac{\partial v_{ij}}{\partial a_{nm}}$ Unfortunately, due to the four-dimensionality a compact notation as for matrix/vector notation is not possible.
However, to make the notation a bit less complicated, the paper uses $\frac{\partial \mathbf U}{\partial a_{nm}}$ But it should be clear what this means.