We moved back to the US at the start of April, so I’m sure the process has changed a bit since then because they weren’t doing any standard appointments then. However, this was our experience.
We had to request an emergency appointment five working days before our flight, providing our reason for travel and flight information. I tried contacting them earlier to avoid a stressful situation days before an international move, but they just sent a standard response.
When I submitted the emergency request form the five days before departure, they did initially respond with some questions, and then they stopped replying to my emails. That evening, I submitted the form again stressing that we were moving due to employment and absolutely needed the appointments. We had a response the next morning and were given appointments for the following Tuesday, which was two days before we left.
Once we had the appointments it was fairly straightforward and the process was very quick. We were out within two hours with emergency passports, and apparently there is no additional charge to apply for new standard passports.
So, it may be a case of gritting your teeth until the last minute and getting an emergency appointment. Although, as I said, this might not be entirely relevant due to changes in the past two months.