Simplifying an upper O-bound in two variables
I assume that $n\ge 2$, which implies $1/(n-1)\le 2/n$. Then

$$ \frac{nm-1}{n-1} \le \frac{nm}{n-1} \le 2  \frac{nm}{n} \le 2m.   $$

Thus your expression is bounded by $O(m)$.