How long does it take to update all nodes that have the blockchain so that they are identical?
The Blockchain is never changed only extended.

As I understand it, nodes do not explicitly check or care whether other nodes have identical copies. They just verify what they receive against what they already have in their copy of the Blockchain.

Synchronisation isn't exactly the right word to describe what happens but it effectively occurs as a consequence of nodes applying the same "consensus" rules. Nodes follow the longest chain, the one with most proof of work.