What time on Oct 15th are extended taxes due? I.e. is it 11:59pm local time if in Central time zone?

It goes by the time zone you are filing from.    Please don't go down to the last possible second----something is almost bound to go wrong if you do that and then you will not be able to e-file.   If you miss the deadline your return has to be printed, signed and mailed.

If you e-file and it is rejected you get a few days to fix it and re-file, but you cannot e-file for the "first" time after the deadline.

If you are getting a refund or do not owe there is no penalty for missing the deadline; you have three years to file.   If you owe tax due you will be subject to late penalties and interest if you miss the deadline.

What if I miss the Oct 15 deadline?

https://ttlc.intuit.com/community/filing-and-paying-taxes/help/what-if-i-missed-the-october-15-filing-deadline/00/25741