It depends on what you mean by "Mongols." Was Chinggis Khan personally aware of France, England, etc.—or even the Mongols more generally during his lifetime? We don't have a ton of sources from the period, so it's tough to say. But shortly after the Mongol invasions of 1241-4, a number of Western European powers tried to open diplomatic relations with the Mongols.

If the Holy See counts as a "Western European nation," three embassies were sent in March or April of 1245. The ambassadors were drawn from a number of different nations—Ascelin of Lombardy was an Italian, Andrew of Longjumeau a Frenchman—and so the Mongols may have had some idea of other Western European nations from these missions, but again, it's tough to say. In 1247-8 the Mongols sent an embassy in return to the Pope, accompanying Ascelin on his return to Lyons. Obviously, this represents the latest possible date the Mongols could have learned about all the countries on the way to Lyons, and they probably had some idea a bit earlier.

By 1248, some Mongols had unambiguously begun diplomatic contact with Western Europe. That year marks the embassy of Eljigidei, a noyan stationed in Persia, to Louis IX of France. I've addressed some of the later (post-1260) relations between the Mongols and the West in another thread, so I'll end this account here, though there were additional embassies and exchanges between 1248 and 1260 as well.

Keep in mind that these are hard boundaries—the Mongols obviously knew about France when they sent embassies to King Louis. In reality, geographical knowledge probably circulated much more quickly, as the Mongols conquered peoples with their own geographical traditions (cf. the geographical traditions of the Islamic, and especially the Persianate, world).

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SOURCES (I'm just gonna copy this from the previous thread):

There's a lot of literature on Mongol-crusader diplomacy, so I'll stick to mentioning Peter Jackson's The Mongols and the West, 1221-1410, which is a pretty comprehensive overview of the subject. It has a chapter summing up much of the previous work on Mongol-crusader relations: "An Ally Against Islam: The Mongols in the Near East" (165-195).

See also (it's a bit older than Jackson's work, but more focused on your question) Denis Sinor's chapter on "The Mongols and Western Europe" in History of the Crusades, ed. Kenneth M. Setton, vol. III (513-544); and Sinor's article "The Mongols in the West" (Journal of Asian History 33[1], 1999: 1-44).