New Year is the first day of the year, which for the Gregorian calendar is January 1. In most countries, that is a holiday.
The sentence shown by the OALD ("We are away over the New Year.") implies "We will return after the New Year." Rephrasing this sentence to use be away, it becomes "We are away until after the New Year." This is not different from the following sentences.
They didn't reach the border until after dark.
He refused to comment until after the trial.
You cannot legally take possession of the property until after the contract is signed.
Until means "up to the point in time or the event mentioned"; in "until after the New Year" the point in time is "after the New Year."