According to this article Wakko's voice is specifically based on Ringo.There's also a touch of Buster Keaton, but you see a lot more of that in Bugs Bunny than Animaniacs.
It's hard to ignore that Wakko's voice isn't a generic British accent, it's specifically imitating one of the Beatles accents. I forget which, I think Ringo but could be wrong. And the style of just approaching and annoying the straight man foils of the comedy world.
https://www.mentalfloss.com/article/30629/way-more-you-ever-wanted-know-about-animaniacs
EDIT: found this, I guess it was John instead