Hungarian entered the Carpathian Basin at the tail end of the 9th century and Finnish and its congeners arrived in the Baltic region around the same time [Ed. some scholars, however, date it to much earlier in the 1st Millennium BCE], ultimately both from the southern Urals (hence Uralic) and Turkey, formerly populated by speakers of numerous Anatolian (ergo, IE) languages, including Lydian, Carian, and Hittite, the first IE tongue, which were overlaid by Turkic speakers from the distant east beginning in the 11th century.
From Language Log.
Note, however, that between the era in which Anatolian language were spoken in what is now Turkey, and the 11th century dominance of the Turkic language, there were also other languages spoken there, including Greek, Persian (which is moderately related to its fellow Indo-Iranian Kurdish languages, which are first attested in the 9th century but are surely at least somewhat older) and Armenian, all of which are also Indo-European languages, as well as the Semitic Arabic language, which was first spoken by significant numbers of people in Anatolia (and also in Iberia) in the 7th century.
Other languages also arose or first appeared in Europe in the Middle Ages.
The Indo-Aryan (i.e. Sanskrit derived) language(s) of the Romani people, a remote branch of the Indo-European language not previously found in Europe arrived in the 9th to 14th centuries. The Semitic influenced Germanic Yiddish language arose roughly the 9th century. The Semitic Maltese language also didn't come into being until the Middle Ages, arising roughly the 11th century.