I can do "English Verse Drama in the 20th Century" in four words: it didn't catch on.
Writers in English tend constantly to hark back to Shakespeare. And early in the 20th century, everyone was making a lot of fuss about the fact that Shakespeare wrote in verse. Everyone got very excited about the meaning and the import of verse, we created the phrase "verse-speaking" (even today, you will hear moronic theatregoers say entirely meaningless things like "they spoke the verse well") and there were a lot of articles written.
T.S. Eliot was the first, in an essay called "The Possibility of a Poetic Drama" in 1920 (linked below), to suggest that verse drama - drama in which the dialogue is written in a specific poetic meter - could be reintroduced. Eliot wrote a whole load of poetic plays, including "Murder in the Cathedral", and "The Family Reunion", that were popular successes. But other writers didn't really follow where Eliot led.
Later - in the 1930s/40s - Christopher Fry wrote a host more plays written in verse, including "The Lady's Not For Burning". They were successful, but again, it didn't catch on.
Even today, the poet Tony Harrison writes plays in verse, most recently a new play called "Fram". I could throw in more names before Eliot (Yeats, Synge) and after (Auden, MacNeice, Beckett) who have written poetic dramas. But, since the 16th century, verse drama has just never found its way back into the main blood of British Theatre.