"There is only one person to blame for this whole situation, and that is me. It's because of my poor judgment that I deserve everything bad that is coming my way," James said.
"This has caused my wife and kids pain and embarrassment beyond comprehension and I am extremely saddened to have brought this on them. I am truly very sorry for the grief I have caused them. I hope one day they can find it in their hearts to forgive me."
In a Warner Bros studio statement, Bullock, who won her first best actress Oscar this month for her role in "The Blind Side," said the trip to Britain had been canceled due to "unforeseen personal reasons."
"I apologize for any inconvenience this may have caused and thank you for your understanding and continued support of the film," she said in a Warner Bros studio statement.
James's alleged mistress, Michelle "Bombshell" McGee, told US gossip magazine In Touch that she had an 11-month affair with Bullock's husband while the actress was in filming for her Oscar-winning performance.
She claimed the pair met up after James responded to her request for information about modeling on his television show "West Coast Choppers."
"He started saying, 'Do you want to hang out?'" and she traveled to Los Angeles to meet him a week later, she said.
McGee said James was ambiguous about his relationship with Bullock.
"We ended up on the couch," she told the magazine. "He wanted to watch movies, but I asked him, 'What's going on with you and Sandra?'"
"He said, 'She doesn't live here. She has a house in Austin. She is filming, and I can't talk about it,'" which McGee said led her to believe the couple was separated.
McGee told the magazine that she and James began sleeping together that night and continued their physical affair for the next five weeks, texting each other several times a day.
Representatives for Bullock declined to comment on the report and James's lawyer could not be reached.
People magazine, another US celebrity gossip publication, reported that Bullock had moved out of the couple's shared home in southern California several days before the In Touch article emerged.
Bullock, 45, and James, 40, met in 2004 on the set of "Monster Garage," a show he presented on the Discovery Channel. They married a year later in July 2005.
The union surprised a number of Hollywood observers because Bullock's reputation for being low-key, discreet and mature contrasted with the bad-boy image cultivated by James.
The tattooed James, a distant relative of the legendary 19th century US outlaw of the same name, had been married twice previously, including to a porn star against whom he fought a series of messy legal battles.
The couple have no children of their own, but care for James's three children from his two previous marriages.
Reports of marital strife came as Bullock celebrated her most successful year yet, receiving a slew of awards for her role as Leigh Anne Tuohy, a feisty southern matriarch who mentors a troubled teenager.
James accompanied her to a series of publicity and award events, applauding as she swept honor after honor.
At the Golden Globes, she dedicated her best actress win to him.
"There is no surprise that my work got better when I met you because I never knew what it felt like for someone to have my back. So, thank you," she said.