CALAIS, France — Serge Orlov, a 62-year-old Briton, likes to rail against what he calls the tyranny of the European Union. Like most supporters of his country’s withdrawal from the bloc, he wants Britain to strike out on its own, a fully sovereign state unshackled from Europe’s pettifogging rules and the Continent’s overweening state.

But faced with excruciating pain and a seemingly endless wait for a knee replacement, Mr. Orlov temporarily shelved his euroskepticism to take advantage of a little known National Health Service program and jump to the head of the line — in France.

After waiting a year just for the possibility of the knee replacement he badly needed, he turned to Calais Hospital in northern France, where in a matter of 10 days he found himself on the operating table for the three-hour procedure, he said in an interview. He plans to get his second knee replaced in a few weeks’ time. Back home, it took him a year to receive a letter informing him when he might have the operation.

“Waiting, it’s just miserable,” he said, describing how he had been shuttled to five different hospitals in Britain over more than eight months. Waiting rooms are “full of sick people,” he said, adding swiftly, by way of explanation, “I can be a grumpy old git.”