HONG KONG — The Hong Kong democracy activist Nathan Law said on Monday that he had applied for political asylum in Britain, a petition that will likely inflame tensions with China if granted. Mr. Law, 27, fled to London in June, just before a new national security law gave Hong Kong sweeping powers to quash dissent after months of fiery anti-government protests. Mr. Law rose to prominence in Hong Kong as a student leader in 2014, calling for freer elections in the semiautonomous Chinese territory through street demonstrations and sit-ins. 