I've recently been reading the book "A History of Hungary", which was written by 18 authors from Hungary, Canada, the U.S., and Western Europe. Slightly outdated (published 1990), it does have a very good over view, written by Laszlo Makkai, on the origins of the Hungarian peoples. While the Finns, Estonians, and others form the region come form the Finn part of the Finno-Ugric proto-culture, the Hungarians and Uighurs came from the Ugric part of the Finno-Ugric proto-culture. Unlike the term 'Indo-European', Finno-Ugric, as far as my knowledge gows, refers not to modern location but to two sub cultural groups. This means that while the Finns and Hungarians are distantly related, the Hungarians are more closely related to the Uighurs. Also, because of the time periods during which the Hungarians lived as part of the Turkic Onogur Tribal Confederacy and under the Khazar Khaganate, they have much more Turkic influence than the Finns. Likewise, they have much more Greek influence becuase of the Byzantine Empire's influence to there south and Roman influence because of the prominence of the Roman Catholic Church in Hungary. While it is true that the Finns and Hungarians are distantly related as members of the Finno-Ugric family, the Hungarians, unlike the Finns, are a Ugric people with much closer ties to the Uighurs, who, I might add, have some words that are similar to the Finno-Ugric words scatered throughout there vocabulary, which many linguists believe, especially when taken in context with other evidence such as folk music and racial type, that the Uighurs originally spoke a Finno-Ugric language but latter adopted a Turkic language do to there circumstances.
But the language is somewhat the same. Post explanations and anything else on this
Highly unlikely on that last one. Uighurs have fairly well been documented through contact with Chinese civilization as speaking a Turkic language as early as the 6th century. The name "Uighur" derives from a Turkic tribal concept of autonomous but cooperating tribes. Also, it is very unlikely that any given people will just give up their native language and start speaking a completely unrelated language. People borrow linguistic terms and concepts, but not languages. If a territory shows evidence of a Ugric language being spoken, then replaced by a Turkic language - the most likely story is that the Turkic-speakers displaced the Ugrian-speakers. 
