Twenty-five people were on board the Embraer 145 jet when the incident occurred at 11.36am on Friday while it was taxiing after landing from Frankfurt, Germany.
There were no injuries and the plane was towed to an aircraft stand, an airport spokeswoman said
Passengers on other flights scheduled to land or take off at Bristol suffered severe disruption as the runway was closed.
The bar worker, who was coming back from a holiday in Krakow, Poland, said: "I'm standing around now with very little direction.
"Apparently buses are coming to take us to Bristol, but that's going to take me another three hours at least.
"I probably made a mistake going on an all-nighter last night."
Friday is one of the busiest days of the year for travel as people fly to spend Christmas with family and friends.
Over 190,000 people are due to travel through Bristol Airport between Friday and January 1, up 5% on the same two-week period 12 months ago.
Dave Joyce, who was due to fly to Javea, Spain, with his family, said his plane was diverted away from the airport.
The 53-year-old finance worker said: "Nothing's moving at the minute and we haven't had any information at all.
"People keep coming but nobody's leaving, so it's all getting pretty crowded just now.
"We've got seats in a restaurant, so we're OK, but my wife's mother is in a wheelchair."
Mr Joyce added: "A great start to the Christmas break ... Javea seems a long way away now."