I think Kazakhstan just buillt a new capital, so in civ/settler analogy, that'd be it. The youngest independent state is East-Timoe, but they should have become independent 30 years ago if it wasn't for Indonesia's invasion. And it had been centuries that the territory of East-Timor was separated from the surrounding lands (Indonesia dutch, East Timor portuguese, and most importantly catholic, which more than anything was the mark of individuality).
If the question refers to the youngest nation-state, which became independent without any previous sentiment of a collective identity separated from the neighbouring countries, that is very hard to say. It would have to be a country born out of political, ideological, economical or geo-strategic reasons, often from the influence of foreign powers. A few candidates in my view would be quite a few african countries born out of the colonial borders, most of them absolutely artificial in terms of previous cultural and political identities, North Korea and some of the archipelagos that belonged to european powers, and wich without any significative self-determination sentiments took the de-colonization ride in the 50s, 60s and 70s and formed a new country. Others, that don't exist anymore would be, in Europe, Yugoslavia and the DDR.
