I want to say Thank you for this information and just want to update for my experience currently (Feb 2022)
My husband successfully passed his interview in January and had his oath yesterday February 9th. We've been hoping to travel to Wales, UK at the end of February and up until this week were just hoping to pay for an expedited passport through a Passport Acceptance Facility and cross our fingers that we got it in 2 weeks versus the previously quoted 3-5 for Expedited Passports. Now, Expedited means 5-7 weeks (from what I can tell the timeframe changed recently - Late Jan/Early Feb). So, we realized the chances of us getting in 2 were practically 0.
So, we have an appointment using the "Urgent Travel" request
As stated above - you must have travel planned for the next 14 days. If you call on a Thur - your travel must be for 13 days from that day - so practically 2 weeks ahead but no later than the Wed. Example - when I called on his oath day Thur February 9th - travel must be no later than Wed Feb 22. They will look for and schedule you for an Urgent Travel appointment at one of the 26 appointment facilities.
Big note - it does not seem nearly as hard to get through as it was above for Original Poster. I called 4 times - once a day before the Oath (at approx 2:00 p.m.?) to test out the phone options, etc and I got through right away. I asked about making an appointment and was given clear information about the two week time frame, etc. I called twice yesterday (at approx 10:20, 10:25 a.m.) and we ended up scheduling an appointment in Dallas as there were none in Chicago (and we can fly direct from our local airport to Dallas on American Airlines), and called back today (at exactly 7 a.m.) and got an appointment in Chicago on Wed Feb 22.
I do know that I had to press 1 for English, 1 for new passport, and 2 for Urgent Passport (and was not able to do a 112 right away to avoid listening to the menus).
Will let you know if he gets his passport the same day as his appointment. I've read some people have to go back the next day and I'm really really hoping this isn't the case, but we will figure it out with travelling to Chicago to get it before our flight to LHR.
Thanks again @Timona for this great post and information.