Prime Minister David Cameron also congratulated the 32-year-old from London, saying he "had scaled one of the great heights of British sporting achievement."
Speaking from his country residence, Cameron added: "The whole country wants to say well done, brilliant, the perfect backdrop and start for the Olympics."
Wiggins had locked up the yellow jersey a day earlier by winning the final time-trial and Sunday's ride onto the Champs-Elysees was largely ceremonial for him.
He also helped Sky teammate Mark Cavendish earn his fourth straight sprint victory.
Many British fans lined the streets of French capital to see Wiggins on his final day of racing.
"We've got the world champion, the Tour de France winner, hopefully the Olympic champion in a week, top of the world," said James How, one of the British supporters.
This 99th Tour will be remembered for successes of other Britons too, like all-rounder Christopher Froome, who was second overall, Cavendish and Scottish veteran David Millar - who won seven stages between them, a Tour record for Britain.