How to resolve the sign issue in a SVD problem?

One does not need to separately compute the eigenvectors of $A A^T$ and also $A^T A$ in order to get an SVD (even in hand calculations). Given an orthonormal eigenbasis for $A^T A$ (resp. $A A^T$), this gives you the right (resp. left) singular vectors. The eigenvalues give you the singular values upon taking square roots. The defining equation for the SVD tells you

$$Av_i=\sigma_i u_i \\ A^T u_i=\sigma_i v_i.$$

This just follows by matrix multiplication:

$$A v_i=U \Sigma V^T v_i = U \Sigma e_i = U \sigma_i e_i = \sigma_i u_i.$$

As an aside, the above pair of equations characterize the SVD through a symmetric eigenproblem not involving $A A^T$ or $A^T A$, which is a crucial step toward developing a numerically stable algorithm for the SVD.

Anyway, if $\sigma_i \neq 0$, to get $u_i$ (resp. $v_i$) it is enough to apply $A$ (resp. $A^T$) to $v_i$ (resp. $u_i$) and divide by $\sigma_i$. If $\sigma_i$ is zero and you want a full SVD then you have some arbitrary choices to make in order to "fill out" $V$ and/or $U$ (specifically you must select any orthonormal basis for the null space of $A$ and/or $A^T$ and add it to the bona fide singular vectors).