Is there a quick way to roughly estimate how quickly a flood will move downstream?
My first thought would be to use Manning equation as an approximation. It does not take into account the effect of a dam burst providing excess water and immediate flooding, although for larger scales (in terms of river reach) this is likely less important. 

Detailed hydrologic models may not be the right answer by the way, especially for the tradeoff of the work involved to build them. A detailed forecasting system with a model and real-time data would be the best, but a quick approximation is often a good guess for the work involved.