As there haven't been any updates in awhile and the embassy has moved, I thought I'd provide our recent experience registering our 19 month old.
We live in Devon and had the "brilliant" idea to take the National Express coach from Exeter to Victoria that morning. At 1AM with a 6:30ish arrival. This should have been a recipe for disaster, but with a nap on the bus and caffeine upon arrival it actually went pretty well. Upon arrival we just bit the bullet and got a black cab. We had loads of time, but didn't want to risk getting lost. It was probably a bit of a needless expense, but at a little over 10 quid it wasn't TOO bad.
I had a passport renewal at 9AM and the baby's appointment was at 9:30. Just across the street from where we were dropped off there is a Waitrose, which was a great place to kill a bit of time. It opens at 7:30AM, so if you arrive before that you'll need to wait around a bit. We had breakfast at the cafe and grabbed a couple of bottled drinks there.
We crossed the street and walked to the embassy around 8:15. This is mentioned often, but there was a massive line outside - this was for non-USCs applying for visas. The line for Americans was practically non-existant. Security was standard federal building/airport stuff - xray machine for bags, walk through metal detector. I had to open and sip from our bottled drinks, but they were allowed.
Once we breezed through security it was on to the 2nd floor. That's the British 2nd floor, so American 3rd. Once you get off the elevator you'll go to an information desk. Our two appointments were combined, we got out number and went to find seats. We hadn't even found a place to sit when our number was called so we went straight to the first window. Handed all of our forms, photos and supporting documents. The lady at the window wasn't over friendly, but she was pleasant enough. This was our one and only hurdle - baby's passport photo was too blurry to use. They have two photobooths though.
We went to the payment window first. Paid in cash as I had enough USD saved from my last visit and from gifts from people back home. For my passport renewal and the little one's registration and passport application it cost a total of $325. When you pay you're given another brief form to fill out for shipping the new passports. We then headed to the photobooths. The first wasn't working at all, the second was but had a card drive that wouldn't work. It costs £8 for a sheet of four pictures and does not give change. You have three chances to take a decent shot per £8. Trying to take a passport photo of a fidgety toddler is not easy to begin with and we did not have anything smaller than a tenner. "Luckily" we got it on the fourth try, as it could've very easily cost way more than the £20 we wound up paying. Might not work for everyone, but we cued up an episode of Furchester Hotel on my Kindle and held it just above the camera. Shee's obviously looking up, but hey, they accepted it! The machine will also tell you if a picture is unacceptable, but tbh, just ignore it. The machine said all of the pics were unacceptable, but the actual humans making the decisions said one of them was fine.
Went back to the first window, handed in the new photos and the address form. We were then directed to another window to pay the shipping fee. Went for the cheapest option which was £8 and a bit. We were given the envelope and directed to wait for our number. This was the first time we actually had to do any waiting the whole time and when we finally noticed the spectacular view of the river. It was really stunning, seriously if anyone reading this goes to the embassy for any reason, take a second to appreciate that view.
Finally we got called to the last window, and this was the best part of the whole experience. The lady we had was so wonderful and sweet. We handed in the envelope and receipt from the previous window. She asked a couple of questions about my time in the US. I didn't have loads of evidence, but my expired passport was enough to prove I could pass on my citizenship. As she was finishing up she said to the little one "Okay sweetie, I'm going to count to 3 and then you'll be the newest American!" Then she counted to 3, pushed a button on her keyboard and said "Yay!" Then the baby threw her hands in the air and yelled "Yay!" so the nice lady left and brought back a little American flag sticker, and the baby threw her hands back in the air and yelled yay again. Maybe they do that for all little kids, but I like to think she was just exceptionally adorable.
And that was it. 3 weeks until we get our new passports. The whole thing took less than two hours, and we probably could have shaved off half an hour if we hadn't had a problem with the photos. We took a cab back to Victoria, it cost the same as it did on the way there. We didn't find any sort of taxi rank, but several drove past, shouldn't have to wait too long for one that is looking for a fair. There is also a bus stop, but not being familiar with the area the timetable was a bit meaningless for us.
So, hopefully this helps someone out!