I don't think so, and here's why. Statistically even today, around 70% of the world's English speakers (not counting those who can't even hold a conversation in the language) speak it as a mother tongue. As a result, they pick up unique local native English idiosyncrasies.
In the case of Esperanto, over 99% speak it as a second language. This means most would learn it formally in school and would tend to use it only in interlinguistic communication. This would promote a normalising effect on the language.
Indonesian is a good example. With over 70% speaking Indonesian as a second language, it is highly normalized among its around 200 million speakers.
Consider too that accent variations if English that existed since before the modern age have been diminishing. So I don't see why Esperanto would break apart, especially due to its grmatical regularity.