Answer:  The Soviet Union.  Their first thermonuclear bomb was air-delivered and truly a strategic and/or tactical weapon.

The US had the first successful thermomnuclear device.  However, “as ultimately constructed, it stood three stories tall, weighed over one million pounds, and used liquid deuterium, a cryogenic fuel. On October 31, 1952 (local time), after the entire atoll had been evacuated, Mike exploded with a yield of 10.4 megatons. The fireball was estimated to be over three miles in diameter. The island of Elugelap disappeared as did portions of other nearby islands. The successful test of Mike ushered in the era of multi-megaton nuclear weapons.

http://www.lanl.gov/history/story.php?story_id=19

“Unlike the first Soviet atomic bomb the development of which was hastened by espionage in the United States, the first Soviet hydrogen bomb was of an original design. In the spring of 1954, the United States tested its own two-stage super-bomb in the Pacific. This type of “deliverable” weapon was replicated by Soviet physicists and first tested on November 22, 1955.”

http://www.soviethistory.org/index.php?page=subject&SubjectID=1954bomb&Year=1954