The only relevant facts, $0=\tau_0 < \tau_1 < \dots < \tau_n= 1$ and $|\omega|=1$ imply

$$\Big|\sum_{i=1}^n\omega^i (\tau_i^m-\tau_{i-1}^m)\Big| \le \sum_{i=1}^n|\tau_i^m-\tau_{i-1}^m|=1\\ . $$ On the other hand, $\Big|\sum_{i=1}^n\omega^i (\tau_i^m-\tau_{i-1}^m)\Big|\to1$ as $m\to+\infty $.

So $1$ is the best possible bound, and the other information is not needed.

The only relevant facts, $0=\tau_0 < \tau_1 < \dots < \tau_n= 1$ and $|\omega|=1$ imply

$$\Big|\sum_{i=1}^n\omega^i (\tau_i^m-\tau_{i-1}^m)\Big| \le \sum_{i=1}^n|\tau_i^m-\tau_{i-1}^m|=1\, . $$ On the other hand, $\Big|\sum_{i=1}^n\omega^i (\tau_i^m-\tau_{i-1}^m)\Big|\to1$ as $m\to+\infty $.

So $1$ is the best possible bound, and the other information is not needed.