Does US diplomats need visa to work in UK?

Across the world, embassies are considered to be territory of the country who's embassy it is.

Thus diplomatic staff working at their embassy are not working in the host country but in their home country.

Diplomats of an embassy would enter a country using a diplomatic passport issued by their country. A diplomatic passport infers on the bearer immunity from many of the laws of the host country (although diplomats are meant to respect the local laws). For instance the embassy would not pay rates and diplomatic cars frequently flout parking restrictions (with impunity).

Thus diplomats would not need a visa to work in the host country but would need a diplomatic passport.