No lower bound: We give an example to show that the expression $\frac{P(B|A)-P(B)}{1-P(B)}$ can be arbitrarily large negative.
Imagine tossing a very unfair coin which has probability of head equal to $10^{-6}$, and therefore probability of tail equal to $1-10^{-6}$.
Let $B$ be the event "tail" and let $A$ the event "head." It is clear that $P(B|A)=0$. Thus our expression is equal to $\frac{0 -(1-10^{-6})}{1-(1-10^{-6})}.$ This simplifies to $-999999$.
By choosing $10^{-66}$ instead of $10^{-6}$, we can make our expression inconceivably huge negative. So there is no universal lower bound for our expression. (If suitable restrictions are put on $B$ and $A$, there may be a lower bound.)
An upper bound: Choose any $A$ such that $P(B|A)=1$, for example choose $A=B$. Then our expression is equal to $1$. It cannot ever be larger than $1$, since for given $P(B)$, the numerator is maximized if $A$ is such that $P(B|A)=1$. Thus $1$ is an upper bound for our expression. This upper bound can be attained, so there is no cheaper universal upper bound.