I'm no expert on English history but I'll wager that Queen Elizabeth I was probably more renowned for her monarching skills than she was for her poetry. A few examples of which are included here and none of which knocked my socks off. It's not clear exactly how much poetry Elizabeth wrote but there was a collection published in 1964 that weighed in at about 100 pages.
George Gascoigne apparently sought to make his way into Elizabeth's court but was never able to quite manage it. He didn't have much luck with farming and at one point even found himself in a debtor's prison. Later he took up the occupation of mercenary and according to one account, he was "a notorious ruffian....an atheist and godless person."
Gascoigne also wrote some plays, one of which served as a source for Shakespeare's The Taming of the Shrew, and he wrote poetry. Norton includes three of his poems. As with Elizabeth, there was nothing here that bowled me over, although And If I Did, What Then? had kind of a nice ring to it.