Don Parrot seems to have done pretty well with hid E-30, but (just a guess here) I suspect you misunderstand the real issues (based on your question)
no camera has appreciable shutter delay. Things vary from 100ms (0.1 of a second) which is perceptible but normally not noticeable) to 8ms (Canon EOS RT and other similar cameras).
However what does get passed around the anals of the internet is the result of the choice to have the camera to refuse to actually release the shutter until AF is confirmed. I get virtually no shutter lag on any camera I've ever owned as long as I focus manually or allow shutter release before AF (though that's unwise for the obvious reasons).
In almost all cases it will come down to lens AF motor type (assuming you have a system with the motors in the lens), focal length (shorter focuses faster) and then availability of vertical contrast (all AF sensors even cross types seem biased in favour of this) and then the camera system.
So, with the 4/3 you'd be wanting SWD lenses to speed up the most critical element in the speed chain.