Long press key to reveal special characters does not work after update to High Sierra
The problem has 'fixed itself'.

Today, I found out the character picker works again. I have no clue why it did not want to work. Possibly, it requires multiple reboots before the functionality is back (I already rebooted a few times in the meantime). Maybe it's because I fiddled with some keyboard settings. I've also used an external keyboard in the meantime, which could (in some strange way) influence the keyboard settings.

I will never know what has definitely solved the problem.

So, if you're having this problem too, then try setting your keyboard layout to that of your locale, but in the layout of your choosing (mine is Dutch language with US international layout). Then reboot multiple times, or maybe turn off your computer completely a few times. If that doesn't work, connect an external keyboard and try if that works.

If you happen to find a solution or even a cause, report back here in an answer or comment, as it will help others facing the same issue.