One of the lines caught my attention. In this version, I think it's "Ev'rybody's in a play," but in the original version on Out of the Blue, it sounds more like "Ev'rybody's in their play." In any case, I realized that this line has some similarity to a verse in the Beatles' "Penny Lane":
Behind the shelter in the middle of a roundabout
A pretty nurse is selling poppies from a tray
And though she feels as if she's in a play
She is anyway
Being in a play (or feeling like it) is common to both. The weather described in "Mr. Blue Sky" ("The sun is shinin' in the sky / There ain't a cloud in sight") and the title itself resemble the "blue suburban skies" in "Penny Lane" too.
I don't know if these similarities were intentional, but there are some obvious and unmistakable references to the Beatles in other ELO songs, so it's possible that "Penny Lane" had some influence on "Mr. Blue Sky."