Is there a plan in place for when the inevitable 32-bit overflow of Unix timestamps occurs?
How much will this impact existing blocks that are timestamped with Unix Epoch times?
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The 32-bit signed integer timestamp runs out in 2038. But Bitcoin uses a 32-bit unsigned integer for the timestamp. That runs out it the year 2106. We'll have to find a solution by then, not by 2038. Since that is still a long ways off, there is no plan for changing the timestamps yet.
Whatever happens would not affect existing blocks. The same thing would happen as with any other fork: blocks prior to the activation block are verified with the old rules, and blocks following activation are verified with the new ones. So old blocks would use the current timestamp stuff and new ones would use whatever new scheme is created.