My experience with the British Embassy is that you have to book an appointment to apply for visa or passport. This is what it says on the website...
Register a birth abroad
You must register your child’s birth according to the regulations in the country where the child was born. They’ll give you a local birth certificate.
This local birth certificate should be accepted in the UK, eg when you apply for a passport or register with a school or doctor. You might need to have it translated and certified if it isn’t in English.

Once you’ve registered locally you may also be able to register the birth with the UK authorities. You can only do this if the child was born on or after 1 January 1983.

You don’t need to register with the UK authorities but it means:

the birth will be recorded with the General Register Offices or at the National Records Office of Scotland
you can order a consular birth registration certificate
You can still apply for a UK passport for your child even if you don’t register the birth in the UK.

Our other members can also add tbeir opinion and advice regarding this matter.
But all in all the service I experienced in UK embassy was very good. They are all straightforward and so friendly. Make sure you turn off your mobile once you get there as a protocol. Thanks and goodluck.