Just replacing crypt32.dll with a newer one didn't change the situation, however what did a noticeable change, was a certain setting in New Moon called "Use OSCP to confirm the current validity of certificates". So now I deactivated the checking of the HTTPS certificate it seems. The error message "SEC_ERROR_OSCP_OLD_RESPONSE" gave me a wink to that. The browser got confused in the process of connecting to the OSCP (holding the certificates), trying to do something with SHA1. Maybe checking the cerfificate? It's not the TLS encryption, SHA is a hash function. I've just learned about this in this video: https://yewtu.be/watch?v=GI790E1JMgw
Some more notes on testing: While I was browsing today on the "next-week" Windows XP machine, none of the HTTPS sites did work, not even msfn.org, swisscows.com and wiby.me. So that changed compared to yesterday. Unchecking the option I described above however made browsing possible again. Still, SHA1 is not repaired... I'd be interested to see someone with an updated Windows XP try to time-travel too (setting the time two weeks to the front), as I am still running SP2 on two XP-computers here. This emergency situation can be simulated! I can confirm, that the time can be set back and SHA1 is recognised again.
The question is of course, if other things break, that rely on SHA1... because what I've described is only an evasion for web browsing with New Moon... watch out for your local power plants!