I am in the middle of the process to get my (first time) UK Passport and I can tell you - it is an involved process. I filled out the form online and then had to wait about 2 weeks for them to print it and send it to me. I then signed and sent it in with the supporting documentation. For a first time adult passport, you must send in your original docs by post (i.e. US passport and, in my case, naturalisation certificate). You cannot get an in-person appointment for a first time adult passport. The documents were returned to me quite quickly/ But then i received a letter that I have to go in for an interview. The earliest date for an interview is 2 weeks away. I know you are talking about childrens' passports, but I can't imagine that the process is easier for first time children passports. I'm about a month into the process and I estimate it will take another 3 -4 weeks.