Firearms in their modern sense start with the matchlock, which came on the scene in the 15th century, so I’ll start there even though the Chinese were using hand cannons as early as the 12th century.
In the early 15th century, we get the matchlock, the first ‘gun’ in the modern sense of the word. It’s quite dangerous to use, actually, but it gets the job done. (I’ve linked to its Wikipedia article so you can see just how risky it looks for yourself.) Its use starts in Eastern Europe and Eurasia and spreads across the continent, and before long everyone is using it despite the obvious risk of burning your own face off with it.
Elizabeth I reigned as monarch from 1558 until 1603. She took over for her half-sister Mary I, who reigned from 1553-1558 (she technically co-reigned with Philip II of Spain, her husband, but she was in charge as the direct heir). Before them, there’s a bit of a dispute, as Edward VI, Henry VIII’s son, named his first cousin once removed, Lady Jane Grey, as his heir presumptive. She reigned for nine days before the Privy Council changed its mind and went with Mary I, and sixteen-year-old Jane was executed.
That’s sort of mangled, so let me put it in a bit of a timeline:
Lady Jane Grey: 10-19 July, 1553
Mary I: 1553-1558
Elizabeth I: 1558-1603
The last female monarch who actually “ruled” England before these three (or two if you don’t count Jane, as many don’t) was actually caught in a civil war between 1135 and 1153 over her right to rule England with her cousin. Her name was Empress Matilda, and she was William the Conqueror’s granddaughter. She was never officially crowned ruler because her cousin usurped the throne whilst she was in Normandy. Although she was a de facto ruler at some point during that period, she is generally not counted as a monarch because she was never officially crowned. In the end, Matilda and her cousin Stephen settled things by naming Matilda’s son Henry II the heir to the throne, thus starting the Plantagenet line. (For more on that whole thing, you might want to read up on The Anarchy.)
…that was a really, really long answer to this question, but basically they did have firearms as we know them today by the time Elizabeth I was in power and the last female monarch to rule before they existed would have been Matilda had she been crowned.