"We are starting to stress a bit now," Mrs Mason said.
The pair paid extra for seats with more leg room, which they did not want to lose.
"We don't want to end up in the middle or not together," Mr Mason said.
After almost an hour's delay there was still no sign of boarding.
"We only have 10 days as it is, just enough time to see the kids," he said.
Some would-be passengers took to social media to vent their frustration.
Victoria Carter tweeted just after 5pm that she gave up queuing to change her flight at Hawke's Bay airport.
The New Zealand Air Line Pilots' Association spokeswoman Lisa Williams said pilots flying at the time of the fault would have lost all radar communication.
"It is an extremely rare event."
Airways New Zealand reported the radar problem was partially fixed at 4pm and was completely operational again at 4.30pm. The backlog was expected to cause further delays.