Situation: Women residing in Spain (but different EU national) gave birth in Spain. Father of the child, who is a UK national is legally known and present on the birth certificate, was present at the birth and the kid took his name as well as the mother's: [name][fathers surname][mothers surname]. Mother and father are not a couple and the father is currently living in England. They currently both wish for the mother to take care of the kid, with the father only visiting on occasion.
Are there any dangers involving the father that the mother should be aware of? For example, does this situation pose any danger for the woman's custody rights over the kid under Spain or EU law? Would it be safe to travel to the UK (or any other country) with the kid, if for instance the father decided to keep the kid in the UK.
Thank you for any advice, and let me know if I should split this into multiple questions.