I know this issue came up a while ago, but while sorting out some documents I found a copy of the procedure and thought it might help:
I'm going to paraphase the document as its a bit long:
After birth and on discharge from the hospital the hospital issues the parents with a copy (white) of the Birth Notification - original is sent directly to the Ministry of Health (MOH) by the hospital.
The father must personally then present himself and the following documentation at the Dept. of Preventative Medicine (old Kuwaiti Hospital). The papers for nationals and non-nationals are slightly different, but I will only deal with non-nationals..
White copy of the Birth Notification
Original passports of the mother and father
Passport copies of the mother and father
Marriage certificate if the fathers name is not mentioned in the mother's passport
This must occur with 14 days of the birth!
A second visit is required to collect the Original Arabic Birth Certificate, it then has to be translated into English and both copies attested by the Department of Preventative Medicine.
Both English and Arabic Certificates then have to be attested at the Ministry of Foreign Affairs (here you can actually send a PRO to do the process)
The birth must then be registered at your embassy or consulate and application made for a passport for the child.
Within 4 months from date of birth the new passport needs to be taken to the Department of Naturalisation and Immigration so that a residency visa can be included in the document.
The Department of Naturalisation and Immigration requires the following documentation:
Sponsor's passport - original and copy (in 99% of cases the father)
Child's passport - original and copy
Child's Birth Certificate - original and copy
Employment certificate or salary confirmation letter (from employer)
Photographs of the child
300 AED for processing the application