What kind of continuation existed between these two time periods?

Pretty much none. After the Ottoman conquest, the nobility was gradually destroyed. Some of the nobility migrated to Hungary, Poland and other Christian lands. Some of the nobility integrated into the Ottoman system of governance (most of sipahis in early decades of Ottoman Bosnia and Serbia were actually Christians, many of them having medieval roots). Some of those who remained accepted Islam, some of them died out; since the beginning of the 17th century there were hardly any left.

Some of the notables in the 19th century did try to forge documents connecting them to medieval noble families. Also, many people tried and forged genealogies during 16th and 17th centuries in attempt to gain nobility status, thus blurring the attempts of historians to determine fates of certain noble families.

Was the nobility of the 19th century mostly a post-independence creation?

Yes, although I must stress that Serbia and Montenegro had no nobility in the 19th century. Serbia had some influental families, but non of them were considered nobility. Montenegro practically had only one noble family - the royal one.

Or were these predominantly people who gained influence under Ottoman rule? Or were these predominantly people who gained influence under Ottoman rule? And what role did they play in their country's wars of independence?

Some of them did. But those truly powerful (Phanariotes, for example), chose to stay in the Ottoman Empire.