In the BA009 incident the only warning the crew got was St Elmo's Fire round the windshield, just before flameout.
There are two problems the engines would see - one is blockage of turbine air film cooling holes due to sand melting in the combustion chamber and re-solidifying on the front edge of the HPT blades, and the other is simple erosion of compressor blades leading to core surge. In either case I doubt you would get much warning.
If the LH and KL test flights are anything to go by the concentration of ash is not as bad as the situation that caused the BA incident. Trouble is, nobody really knows.