Nationality law. For example, the British Nationality Act of 1948 establishes worldwide jurisdiction in the matter of British nationality. Obama was born under the U.K.'s nationality jurisdiction.
Much like the U.S. has established worldwide jurisdiction over its citizens in the matter of being required to pay federal taxes on their income regardless of source.
And unfortunately for you - American law prevails for those born in America. They can exercise their British citizenship - but in situations where you have dual citizenship - the American law prevails. Every time.