Re the Air India crash - yes the 'duty' Chinook was airborne from Odiham two hours after the incident (getting the winch on and overload tanks in took the time), but we weren't the first; an ad hoc crew grabbed from the mess/quarters was off very quickly . However, the greater part of the work was done by the RN and RAF Sea Kings; even the CH 53s with AAR Hercs from Woodbridge appeared on Day 2.
As far as 'Aid to the Civil Community' goes, I have noticed that over the years it has become less and less as the Treasury have turned the screws on those very communities requiring assistance. It seems to need higher and higher approval, and then the community requiring help is charged ever more exorbitant sums (to cover the costs of running the military it seems), Most of the crews would like to be involved, and the aircraft are there, but the arcane rules on costs (after all it is now a budget run system!) seem to prevent it.