Hello,
I just registered a birth recently. It's very easy. At the hospital you get a certificate/confirmation of the birth which you bring to the civil status office on Mahe.
You need your ID/Passport and if registering the father as well, he needs to be there and bring his identification with him too. You also need 2 witnesses for the act. The whole thing takes about 10-15 min. I think you need to do this up to 21 days after birth, but it's best you get it done when leaving the hospital.
You then get a handwritten birth certificate. With this, you can get the ID Card for your baby if one of the parents are Seychellois or if the parents are Residents. Most embassys / consulates need the A4 version though which you need to request separately. You will need to bring a 50 Rupee stamp with you.
I registered the birth of my child in my home country through the consulate. They told me which documents I need. They did the rest. Because the father of my child is Seychellois, it now has dual citizenship. But if both parents are non-Seychellois the child will NOT just get citizenship of Seychelles through the act of birth.
Cheers!