"to face" and "to face up to" are different verbs. Your dictionary certainly has separate definitions.

"to face something" is to approach it head on / "to face up to something" is to admit guilt or wrongdoing.

In the context of "they face delays of up to 2 months", "up to" simply means "as much as" or "maximum". They're trying to say that delays can last, at the most, 2 months.