Some time related things have stopped working for me:

1. My settlement does nothing. No more decisions, nothing to build, just sitting there and stacking up debit.
2. Ares and Helios always only tell me to come back later
3. The "survive in extreme environments" milestone is stuck at 5.2 sols.
4. The console operators in my base don't hand out any more quests

What can I do to make time run again?

The "survive in extreme environments" milestone has been stuck for me for many many months. I would also like to know how to get it moving again.

With the sentinel update, problem 1 seems to be fixed.
The other ones still persist though.

For 2.:
If you play in Single Player Offline-Mode, this is normal, the interactions with both is only available with network connection.

For 3.:
The time for the milestone count only the time in extreme conditions, the description say it. If you explore on planets with extreme weather, the timer goes away, otherwise he stay.

for 4.:
If the main first missions ended, you can send the boys to Florida. They stay around and doing nothing for the game again. I have they placed in bases, where they are as decoration. Build her terminals somewhere at you opinion and call them (on PC with "E") and they stay on this terminal so long, till you call them to another terminal.