The American embassy (in London) used to be notorious for refusing to pay any parking tickets of congestion zone fees. Didn’t wanna, can’t make me is the attitude.
Foreign embassies and diplomats are exempt from the taxes of the localities where they are situated - this is a long established principle.
The US claim the congestion charge is a tax, so they are exempt, and you can't serve papers on them for debt under the Vienna Convention.
(That said, if the congestion charge were enforced with barriers and gates, they would have no choice but to pay, and then work out how to claim it back from HMG).