Is It Correct to Refer to London as the UK?

No. London is a city; the UK is a sovereign country. London is the capital city of England and the de facto capital of the United Kingdom. The United Kingdom of Great Britain and Northern Ireland comprises four constituent countries: England, Scotland, Wales, and Northern Ireland.

Referring to London as "the UK" is a category error—like calling Washington, D.C. "the United States" or Paris "France." It conflates a part (the capital city) with the whole (the nation-state). While London dominates the UK economically, politically, and culturally, it is only one of 69 official cities in the UK and represents about 13% of the national population.