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According to the British Nationality Act of 1948 Barack was born to a British subject and therefore became a British subject at birth. Don't take my word for it you can read right from the Obama campaign! (no longer there but was for years)

“When Barack Obama Jr. was born on Aug. 4,1961, in Honolulu, Kenya was a British colony, still part of the United Kingdom’s dwindling empire. As a Kenyan native, Barack Obama Sr. was a British subject whose citizenship status was governed by The British Nationality Act of 1948. That same act governed the status of Obama Sr.‘s children.
Since Sen. Obama has neither renounced his U.S. citizenship nor sworn an oath of allegiance to Kenya, his Kenyan citizenship automatically expired on Aug. 4,1982.”

Sneaky....The Kenyan constitution does state that if you do not denounce your other citizenship and swear an oath to Kenya then you citizenship does in fact expire. (Don't pay attention to the fact that he even was a foreign Citizen...) But they give you a two year grace period so the date Aug 4, 1982 should read Aug 4, 1984. Big deal Greg...Well it is. Enter the British Nationality Act of 1981.

Every person who—
(a)under [F105the British Nationality Acts 1981 and 1983][F106or the British Overseas Territories Act 2002] is a British citizen, a [F107British overseas territories citizen] , [F108a British National (Overseas),] a British Overseas citizen or a British subject; or
(b)under any enactment for the time being in force in any country mentioned in Schedule 3 is a citizen of that country, shall have the status of a Commonwealth citizen.

The UK was terminating the title of "subject" and replacing it with "Commonwealth Citizen" in this act. So the date above is relevant because the commencement date of the BNA 1981 was Jan 1, 1983 while Obama was still considered a Kenyan Citizen thus being a British Subject then became a "Commonwealth Citizen" of the UK.