The process is the same for a first child or adult passport, but you will have to attend an interview in person and provide identification as other supporting documentation.  These interviews are often held at your local British Consulate.
Further information on UK passports can be found here
Supporting Documentation  
Vital: When you are making use of for a primary youngster or adult passport the method is fairly equivalent, Despite the fact that you’ll have to show up at an interview in individual and provide identification as well as other supporting documentation. Interviews are held as close as feasible to your house. 
The procedure is identical for a primary youngster or adult passport, but you’ll have to go to an interview in person and supply identification as other supporting documentation. These interviews will often be held at your local British Consulate.
Even further information on United kingdom passports are available in this article
Supporting Documentation 
Your current Primary (copies not appropriate) ten calendar year Grownup or five year youngsters┬┤s passport.
Two passport photographs of on your own of suitable measurement. To ensure that the images are acceptable, you should Adhere to the IPS rules, a hyperlink to which you’ll locate underneath. You have to make certain the images are in colour; equivalent sizing (45mm x 35mm), and of your head only, without any hat or headdress, your facial expression should be neutral (no smiling, grimacing, or or else pulling a face).