"to face" and "to face up to" are different verbs. Your dictionary certainly has separate definitions.
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"to face something" is to approach it head on / "to face up to something" is to admit guilt or wrongdoing.
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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.
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