This is as good as I can find with Google's assistance:
As for “swimmingly,” it means to do something easily or successfully. It’s known from the early 1600s.
Today, it’s typically used in the formation, “It went swimmingly.” For example, “My unclaimed luggage column did not go swimmingly.”
“Swimming,” as an adjective, has also been used to mean the same thing, though not for so long.
The usage seems a little weird. After all, swimming is anything but easy, at least for humans, even highly trained ones.
But it’s really got more to do with how easy swimming looks, especially for fish and other aquatic life. It all goes back to the oldest adjectival meaning of “swimming”—something that swims by nature.