laurainspain,
your baby is both spanish citizen and british citizen at birth because both citizenship are acquired “jus sanguinis” that means by blood.
you are required to register the birth of your children born in spain at the local registro civil and will be registered as a spaniard, your husband should know this. primarily, for documentation because when you travel by air with baby, the baby would need a passport or to be included in the libro de familia.
you can also go to the british consulate later and they will inform you how to register the baby so he could also have a british passport, in case you want to travel with your baby using a british passport.
while here in spain, your child can go on using spanish dni. if he or she travels, he/she can use either passport depending on convenience.
i remember many years ago, a plane was hijacked. the hijackers released all passengers holding filipino passports (some held usa passports but switched to filipino passports so they could escape with the rest)
we hope this (hijacking) never happens again but we never know. if your children hold more than one passport while traveling, this could be an ace up their sleeves.
