But do Americans understand cheeky cockney chappies, or for that matter what has come to be called "Estuary English"

I recognized the accent, even if I didn't have a name for it, but is there something specific about ``cheeky cockney chappies'' that you're referring to, or just a general notion of what they're like?

I think he has the most irritating use of an English accent since Dick Van Dyke in Mary Poppins and I don't believe it's his given tongue.

How is more irritating than anyone else using it? Is it because you believe it's an affected accent (rather than English not being his native language, which is what I would take your opinion to be from the latter part of that sentence)?