I love the idea of this thread, and the idea that there should be a different answer for different parts of Europe, like 1485 for Britain, 1492 for Spain etc...
I generally considered the middle ages to end when Spanish exploration and the Colombian exchange began, which is a good broad, continent spanning definition.
I am intrigued by this Long Middle Ages idea. From a US perspective, the statement "In Mississippi, the Middle Ages ended in the 1960's." is not entirely crazy. From a Western European perspective, this would probably have the Middle Ages ending during the French Revolution->Napoleon->UK Reform Acts->Italian and German unification period. This would also be around the time economic growth was obvious to the average person in Europe.
In the US, the generalized end of the "Long Middle Ages" would be the 1860's.
Nathan already pointed this out very well, but the idea of the middle ages ending so late in the US is pretty puzzling. Most of the society defining traits of the Colonies/U.S are products of a post-middle ages world in Europe. Although the tidewater colonies used indentured servitude initially (both Europeans and Africans), after a few decades that was phased out in favor of chattel slavery which, according to Colin Woodward-was imported from the British Caribbean colony of Barbados. Plus, you have the transition from a Mercantile economy to a Capitalist one and the aristocracy/rigid class system that Europe had (and waned over the 18th and 19th) was nonexistent in the North and frontier, of course the deep south/tidewater had a de-facto aristocracy all it's own.
Basically greater European society was altered permanently when colonization and the Colombian exchange ramped up in the early-mid 16th century. The U.S. never really resembled pre-1500 Europe. And everything Nathan and Antonio V said about the rapid changes a time-traveler would notice.