Changeling said:
When you have possibly done something that is somewhat incorrect like leaving a polishing compound on to long and it has dried!
Spray on some Meg's #34, let it dwell for a bit to soften the polish, wipe off with a plush MF. Try to use polishes that don't stain too badly.
..Getting a polish, glaze, etc, on the rubber or plastic trim and it stains!
If a solvent or APC won't get it, then I use "Dried Wax Remover" from Griot's. Others undoubtedly sell the same stuff cheaper. But honestly, I wipe it off right away (#34 or Hi-Temp's Remove-All on a MF) before it dries and can't realistically imagine a situation where I'd be unable to do that. And again, I simply don't use products that are known for awful trim staining.
..Having to leave a carnuba on the vehicle overnight (whatever) because of an emergency!
At the risk of sounding flippant, I'd never apply any LSP so thickly that an extended cure time would be a problem (note that I don't use Victoria's, which IIRC could cause problems if left on too long). Even waxes like Souveran that you're "supposed to" (scare-quotes intentional

) W-O-W-O come off easily if you don't put too much on.
.. Before you can polish off your wax job there is a sudden downpour, windstorm, hurricane, etc,..
I'd just wait until conditions stabilize and rewash the vehicle, then inspect it and go from there (worst case: repolish and then wax). But I'd always know about local weather conditions before I started detailing (or doing anything else).
...[i.e.,] How do you approach these calamities/emergences with some sense of intelligence, what products, methods, skills, action do you take other than saying holy **** now what do I do..
Same way I ensure that such things don't bite me in other areas of my life

Research (such as your asking these Qs), prioritizing, situational awareness, foresight concerning the potential consequences of my actions, and preparedness with regard to how to deal with those consequences.
I ensure that an emergent event won't cause problems by never letting things get to the point where they're beyond my immediate control/rectification and a *huge* part of this is simply (effective, focused) thinking. Polish on trim? Notice it, wipe it off; likelihood of bad weather? Know about it and plan for it; chance that you won't be able to remove the LSP? Apply it correctly (thin) so it won't matter. I've had serious situations arise while detailing that required me to stop work *immediately*. The worst thing that ever resulted was water spotting from having to stop in the middle of a wash (which was nothing compared to the medical emergency at hand).