I haven't given birth abroad yet but am due to in the next 4-6 weeks...
First I will need to register the birth within 3 days with the swiss authorities and will get a swiss birth certificate for the baby. All countries I assume will ahve different regulations for this but here I need both our passports, both our residence permits and official copies of our birth certficates that are less than 6 months old. I also need a copy of our marriage certifcate. For unmarried couples the father should sign an official, witnessed recognition of paternity document at the local commune offices.
Once we have the swiss birth certificate we can apply for a british passport for bub because we both have full british passports.
Within the first year once everything has settles down we will also register the birth with the british consulate so that bub has a british birth certificate too, this should make things easier for him/her in future although isn't a legal requirement. Here is the link to the government site with info on that.
http://www.fco.gov.uk/en/travelling-...register-birth
Hope some of this info helps.