We had our Consular Birth Abroad/Passport/Social Security appointment last week in London, so wanted to give a recent update for anyone else going soon.
Our appointment was for the 10:30am slot. We arrived around 10am assuming there would be a line to go through security, but while the line for NON-US citizens was long, the US Citizens line had nobody so we were able to go right through. We had our 3 year old and the newborn with us and had no problems bringing in snacks, stroller, etc. There are lots of stairs and no lifts, so be prepared to lift the stroller up and down stairs as needed. When we walked in, a man checked our paperwork, said it looked good and then we were able to pay for the courier service if needed. After that we went up more stairs the waiting room. There were TVs there (one with kid's show and the other was CNN) and there is also a soft play area for toddlers with other puzzles/toys. Changing room/toilet is down 2 flights of stairs but there is a water fountain. We didn't bring any electronics with us (left it all at our hotel) so I don't know if they would've held anything for us if we had accidently brought it.
When they call yoru number you go to the window to give your paperwork. Then you go to another window to pay for the services, and finally, they call your number to meet with the officer and get the approval. At the first window, they specifically asked for our transcripts (which I almost didn't bring because others said not needed, but printed them last minute). I am sure it would've been fine with other documentation we had, but it was nice we had them (college ones) since they asked specifically for them. Overall, the whole process was streamlined and easy. We enjoyed talking to the other Americans with little ones about life int he UK. The whole process took us an hour and a half and we were out the door. Easy Peasy
She said to expect the passport in 2 weeks and social security in 6-8 weeks.