Is it possible to obtain the value committed to by Pedersen commitment given the blinding key?
Recovering $v$ from $\alpha$, $G$, $H$ and $P=\alpha\cdot G + v\cdot H$ requires solving the discrete log problem to find $v$. Specifically, note that $v\cdot H = P - \alpha\cdot G$ so you need to find the discrete log of this value with base $H$. As such, if $v$ is small then you can find it.

It is crucial to understand that although it is hard to find $v$ in general, it does not mean that revealing $\alpha$ is OK and $v$ is still securely committed. This is because some part of $v$ may be revealed by $v\cdot H$. Thus, this is neither a secure commitment, nor possible to obtain $v$ efficiently.

On the note of working modulo $p$ or over Elliptic curves, this makes no different essentially.