WebP is mostly handled via transparent proxies - meaning people with proper browsers and Android mobile devices get WebP (large market share), everyone else gets (larger) PNGs.
That way you don't really care either way on the backend.
Saying 'proper browsers' gives away your bias there, since only the Blink WebKit engine supports it (Chrome/New Opera), which is under 20% of the desktop market worldwide.
That way you don't really care either way on the backend.