I'm in a bind here! So I started the part of You Know It When It Happens where you're supposed to search for clues so you can protect President Kimball when he arrives.
I saved my game, quit and went to make my dinner.

When I opened the game again and loaded my save game, it had simply skipped to a stage here the president is already dead. This is the case for both my autosave and my quicksave.
My last regular save is an hour before, and while I technically could go back and play from that, it seems an unnecessary thing to do because of a bug.

Does someone know how I can reset the quest with a console command or some similar solution?

Try this I've never played on PC but I think you just need the quest ID and the command to reset it.

You can't do it with *a* console command. The way I see it you need to reset the quest, set stages (preferably one-by-one), resurrect Kimball, set variables right so he restarts his speech, you need to check if the assassins are still there..... A looooot of stuff if you want it right. Of course if you just want to directly reward yourself for saving the Prez, just reset the quest and set stage to complete the game.. But in game world Kimball dies and people will likely say that he died.