You book an appointment via UPS, but it takes some days before you find a slot and then they give you up to 6 weeks before you receive the new passport, if you opt for the (expensive) expedite option. So in the end we decided not to renew the expired passports, as we were leaving 5 weeks after a potential appointment. I researched the issue and it seems fine to travel back with an UK/EU passport, although in theory you should not do that. Apparently, they can't force you to remain in the USA and they do not fine you for that. My wife also called the border police at San Francisco airport and was told they do not mind. I have to say the whole thing make me nervous. We are flying back on Friday, so I should be able to report back early next week. Fingers crossed it will work just fine.
Good luck,
Matteo