Yup, deal with actively vomiting first by grabbing bin and thrusting under chin, then dragging/carrying/propelling bin and DC to bathroom to use the toilet to vomit into.
Parent on DC duty then soothes DC and mops them up. Clean first, poss shower or bath if needed, then clean pjs/clothes and drink water. Then any further snuggles, pain relief etc needed before back to bed.
(OrYes we're often middle of night with 2 parents available).
Other parent deals with mess - whip sheets from bed before soaking in happens, then floor, other surfaces, then clean sheets. Then put dirty sheets in wash and get fresh water for child.
Parent on child duty could help clean up if bath was needed once child was old enough. Also if it was fast and not much cleaning of child needed, both of us would do room once child sorted to get back to bed, then both deal with aftermath outside Troon (clean bathroom, sheets in wash etc) once child back in bed.
When solo parenting, priority was get child to stop vomiting then settle/clean them, then clean up mess as soon as possible before it settled into anything soft (carpets, bedding, sofa) and only then - subject to child settling, doing harder surfaces cleaning, putting in wash, emptying bins etc.