A combination of emergency and "tly" less the "cy". Emergently is an additional expletive for emergency, something that it is "really an emergency of importance!" (a true emergency, commonly used in medical vernacular). adverb
The word emergently is commonly used in hospital settings such as: He had to "emergently" leave town due to the impending death of his girl friend, mother, lover, other sibling, in-law, or child of importance.