The short answer: Both.

Eliot was born in Middle America (St. Louis), into a prestigious Boston family that had produced two U.S. presidents. He was educated at Harvard, and strongly influenced by the natural environments of both the Mississippi River and coastal New England.

In 1914 he moved to England and married an Englishwoman, and in 1927 he joined the Church of England and became a British subject. Over the years he developed a "BBC accent" and an "English" demeanor (although some people insisted that he would only ever be a caricature of an Englishman), and lived out his life in London.

Eliot's American roots and his love of England are both apparent in his work, and fused in his later poetry. FOUR QUARTETS navigates his reverse immigration from America to England, without ever losing sight of his childhood home. In a 1959 interview Eliot explained that his poetry "wouldn't be what it is if I'd been born in England, and it wouldn't be what it is if I'd stayed in America."

To make this answer even more non-committal, let me add that Eliot can also be viewed as a European poet. In 1918 he wrote (in an essay on Henry James) that "it is the final perfection, the consummation of an American to become, not an Englishman, but a European." For most of his life, Eliot regarded Dante as the ideal poet.