In a recent post:
the discussion established that in English poetry the authors which comes after Shakespeare are Chaucer, (+/- Spenser), Milton and Wordsworth.
I'd like to do the same now for prose and drama.
1. English Prose: This is easy - there are only two books of prose with comparable impact to Shakespeare: the Book of Common Prayer and the Authorised Version of the Bible (King James Bible).
So, the second-best prose writers to Shakespeare in English Prose are Thomas Cranmer, William Tyndale, Myles Coverdale and various other collaborators.
But if the fact that the second greatest writers of English prose are translators and a (kind-of) committee (albeit divinely inspired) is hard to stomach, then it gets much harder to identify a second placed writer.
I would put forward Samuel Johnson.
2. English Drama: This is easy - aside from Shakespeare there is only one dramatist in the English language who is productive and quotable enough to stand near him: George Bernard Shaw.