Your first sentence is another way of saying:
The first version of the original, written in Latin (the language of science at the time), has not survived until our time; neither has the second [version].
In this formulation, there is no question but that 'has' is correct. (You could use 'nor' in place of 'neither'; in fact, my first version of this text did, but it didn't survive until the time I hit submit.)
On that basis, I think that 'has' is also better for your 'neither/nor' version, though I agree with FumbleFingers that 'have' is not traumatically wrong and few would object to it, even if the strictest of grammarians could object to it.
In your second sentence with 'versions', there is no question that you need 'have'. If you extend the list of cited versions:
Neither the first nor the second, fourth or fifth versions of the original, written in Latin (the language of science at the time) have survived until our time[, but fortunately a battered copy of the third edition shows...]
So, if you use the singular 'version', use 'has'; if you use the plural 'versions', use 'have'.