The most difficult part of the process (not dealt with in this version of Passport Application but maybe a future DLC pack?) was actually finding someone who could certify my evidence (you are meant to submit originals but they keep the docs including passports for 3-6 months which is a bit unrealistic if you are living abroad). I can't remember the exact rules but it wasn't possible to use a US notary or a normal solicitor certification process and instead I needed to go to a council office.
After calling about 5 councils all of whom disavowed any knowledge of the process or its requirements I ended up finding someone at Islington Council who was delightfully helpful. But it was one of the more frustrating UK government interactions I've had.