Arguably, the simplest case is where the matrix $\bf A$ is symmetric and positive semidefinite (PSD) and ${\bf u} = {\bf v}$, which ensures that the eigenvalues of the rank-$1$ update are in $\Bbb R_0^+$. Hence,
$$ \begin{array}{ll} \underset {t \in \Bbb R} {\text{minimize}} & t \\ \text{subject to} & {\bf A} + {\bf u} {\bf u}^\top \preceq t \, {\bf I}_n \end{array} $$
which, via the Schur complement, can be rewritten as the following semidefinite program (SDP)
$$ \begin{array}{ll} \underset {t \in \Bbb R} {\text{minimize}} & t \\ \text{subject to} & \begin{bmatrix} t \, {\bf I}_n - {\bf A} & {\bf u}\\ {\bf u}^\top & 1 \end{bmatrix} \succeq {\bf O}_{n+1} \end{array} $$
This is not very satisfactory, but it may be better than nothing.