Without mistaking the Mongol Empire as having been the contemporary Tatars, Mughals, or Qing Dynasty
None would have known precisely, some would have known with some imprecision, some wouldn't have known at all.
Interestingly, there would have been _more_ awareness at earlier dates than you suggest; the Mongols were no longer "great khans" by the 17th century, but in the time of their greatest expansion, they were known in Europe and England-- though most often written in languages other than English.
As regards the Mongols, our knowledge of their history, of their customs, of their way of life, our relations with them, England presents an interesting case. We do not know the extent of the material lost on the Continent, but, in this (for the Mongols) remote corner of Europe, (in places safe from their devastation) documentation is to be found. A monk of Saint Albans, the chronicler Matthew Paris who died in 1259, is an important source. He was the only person to preserve Ivo of Narbonne's confession (which reveals that an Englishman was one of the first envoys of the Mongols to King Bela of Hungary), the report of Bishop Peter of Russia given at the council of Lyons in 1245 and information about Andre of Longjumeau's mission after the council.
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Matthew Paris's death marks a decline in the interest of the chroniclers in the Mongols.
The Travels of Marco Polo were well known in Europe-- from long before the Mughals or Qing, who were not contemporaries of the Mongols. Polo's travel to the court of Kublai Khan took place in the late 13th century, and the first editions we have are from the early 14th, written in Old French. The first English language version was published in 1503 - a translation of a Spanish version-- by John Frampton "The most noble and famous travels of Marco Polo". This English edition does not seem to have been widely read, but those who did read it, or had read more popular editions in continental languages, would have aware of the Mongols.
In European minds the line between Mongol and Tatar would have been vague, because Tatars are Mongols-- a contemporary reader should not have confused the Great Khan with dynasties that come hundreds of years later: the Qing (dynasty begins 1636) or the Mughals (dynasty begins 1526)
There would have been other potential sources for an Englishman to become aware of the Mongols. England carried on substantial trade with Russia, and the Russians were clearly aware of who the Mongols/Tatars were, though again -- they wouldn't have confused them with Mughals or Qing. For later Englishmen, the expression "cruel as a Ta[r]tar" would be familiar, an epithet that mostly reflects Russian historical experience with the Golden Horde.
If you're thinking of perhaps the most celebrated reference to a "Mongol" subject in English letter, Samuel Taylor Coleridge's poem "Kubla Khan" [completed 1797, published 1816], there's nothing in it beyond oriental fantasy-- no sign that he knew anything beyond the name . . . its a poem more motivated by laudanum dreams than history.
Sources:
The legacy of John Frampton: Elizabethan trader and translator
"Know Thy Enemy": Medieval Russian Familiarity with the Mongols of the Golden Horde
England and the Mongols (c. 1260-1330)
A New, Probable Source for "Kubla Khan" [of Coleridge]