Just an update on this:
I mailed out the application for my new passport in my married name on 7 February, via
Royal Mail Special Delivery. The U.S. Embassy received it on 8 February. On 9 February, they charged my credit card with the $67.00 fee. I received my new passport today, 1 March. The "issue date" on the new passport is 14 February, so they actually did it pretty quickly. They really lived-up to their 15-day claim. Yay!
Also, included with my new passport, they include a bunch of helpful leaflets: how to transfer my British residence permit, how to apply for an absentee voter ballot, and how to register with the U.S. Embassy.