This photo looks like it was taken with a fisheye lens (it's a great photo, by the way).
The thing is, a shot with a fisheye lens is a poor example of your point about lag time. Mirrorless cameras have longer focus lag than dSLRs, but there's an easy trick to avoid focus lag with a fisheye lens: switch to manual focus. Most fisheye lenses stay in focus from about a foot to infinity, and that extreme DOF makes it easy to take photos without resorting to autofocus.
Once you eliminate focus lag, there's no difference in speed between mirrorless and SLR cameras.