Weird - all of a sudden the menu now works for me again!
After not having rebooted (just leaving the laptop on and using it daily), and not changing any settings, the menu now appears as normal!
I also noticed that in TextEdit, there is a weird "conflict" if you are trying to accent the first character in a sentence (or paragraph). In that case, holding only the vowel key (e.g. 'a' - without shift or caps lock) will first produce the auto-correct pop-up after 1/2s, with only the capitalized letter (with a small closing 'x' to the right of the capitalized letter), but then after another 1s, the auto-correct pop-up is replaced by the accent menu. I have not noticed that before, but it make sense (and I can live with it).