All three are quite interchangeable. The difference between plus and davantage is mostly formality and has already been answered here:
Davantage/plus – what's the difference?.

Encore might suggest that’s not the first time extra time has been needed for the task.

Note that both plus and davantage use de temps but encore uses du temps:

Il me faudrait plus de temps

Il me faudrait davantage de temps

Il me faudrait encore du temps

See also: « Une fois de plus » et « encore une fois »

“Plus de X” and “davantage de X” (where X is a noun without an article) are completely synonymous. “Plus de X” is more common. You may occasionally find people who forbid “davantage de/que” and only allow davantage as a standalone adverb, but even the very conservative Littré allows them. Note that “plus de” is pronounced [plys]: the S is not silent when plus means more.

“Encore du temps” is almost synonymous, but there is a subtle difference which may or may not make it inapplicable. I perceive the difference intuitively as a native speaker and I’ll try to explain but I’m not completely sure I’ve captured the difference correctly. The meaning of “encore” always includes some nuance of “again” and not just “more”. It can mean adding more to what’s already there, but not just adding more to a certain quantity. In your first sentence, if you’re currently giving a presentation and want to say that you cannot go into details because you’ve run out of time, you can say either “il me faudrait plus de temps” or “il me faudrait encore du temps”. But if you’re preparing a presentation and you decide to cut a topic because you don’t have enough time, only “il me faudrait plus de temps” works.

This is not specific to time. For example, encore works in the first context below but not in the second context:

Je sens que la pâte est trop dure. Il faudrait plus d’eau.   (I can feel that the dough is too hard. It needs more water.) [Context: while baking, I’ve already put some water.]

Je sens que la pâte est trop dure. Il faudrait encore de l’eau.   (I can feel that the dough is too hard. It needs more water.) [Context: while baking, I’ve already put some water.]

Un verre d’eau ne suffit pas pour cette recette. Il faut plus d’eau.   (One glass of water is not enough for this recipe. It needs more water.) [Context: discussing a recipe.]