Proving that English is not a regular language
It is not easy to prove the complexities of natural languages. See Complexity of natural languages for some hint that English language is not regular.

In fact in Evidence Against the Context-Freeness of Natural Language,
Stuart M. Shieber shows that many natural languages are not even context-free.

However, I only know about Dutch and a dialect of Swiss German, for which proof of non context-freeness exists. And the proofs are quite old, the results about them were proved in 1980s.

For detail you can also refer - The Handbook of Computational Linguistics and Natural Language Processing
edited by Alexander Clark, Chris Fox, Shalom Lappin.