“We were due to board at 17:00 BST but nobody turned up at the gate to let us through,” Robertson said.
“My husband who is in the UK texted to tell me that there was an issue with air traffic control.
“So far we have been told we would be delayed for two hours hours, then six hours – my parents are 90 and 88.”
Lisa Jacobson, 50, said her daughter was due to fly from Bristol to Faro at 13:30, but had been grounded for more than five hours.
Others took to social media to report being stuck overseas, including in Crete and Venice.
In its initial statement, Bristol Airport said the issue was only impacting departures, before clarifying arrivals were also being delayed.
“We recommend passengers travelling today to contact their airline directly to check the latest flight information and prior to leaving for the airport,” a spokesperson said.
A spokesperson for Nats said: “We have identified that the issue was in our flight processing system. Our engineers are on site and working urgently to resume normal operations as quickly as possible.
“Flights are still departing and airspace is open, but recovery will take some time.”