I have a matrix $A \in \mathbb{C}^{n \times n}$ in the form $A = uv^T + B$, where $u,v \in \mathbb{C}^n$ and $B \in \mathbb{C}^{n \times n}$. I know $A$ is invertible and want to find an upper bound for the norm of its inverse using the vectors $u,v$ and the matrix $B$. More precisely, I want a bound for
$$\| A^{-1} \| = \| (uv^T + B)^{-1} \|,$$
where the norm can be the spectral or the Frobenius norm.
I'm trying to come up with something in terms of $u,v,B$, but with no success. Also, searching here I found some posts about sum of psd matrices (like here and here). This is not my case, unfortunately, but I still hope there is some idea out there to help me on this problem. The matrices $u^Tv$ and $B$ may be assumed to be invertible too.
EDIT: In fact, I was able to get a bound, it is close but still not exactly what I need. Using the Sherman-Morrison formula we get $$\|(uv^T + B)^{-1}\| = \left\|B^{-1} - \frac{B^{-1}uv^TB^{-1}}{1+v^TB^{-1}u}\right\| \leq \|B^{-1}\| + \left\|\frac{B^{-1}uv^TB^{-1}}{1+v^TB^{-1}u}\right\| \leq $$ $$\leq \|B^{-1}\| + \frac{\|B^{-1}\| \|u\| \|v^T\| \|B^{-1}\|}{|1+v^TB^{-1}u|} = \|B^{-1}\| \left( 1+ \frac{\|B^{-1}\| \|u\| \|v^T\|}{|1+v^TB^{-1}u|} \right).$$
In order to have something useful for my purposes I need throw away or bound the factor $v^TB^{-1}u$ in the denominator. Suppose we have the norms of $u,v,B,B^{-1}$. Is there a way to use this and make the factor $v^TB^{-1}u$ disappear (I don't want to compute the inverse, only the norm of the inverse is accessible).
Thank you.