You just need either a letter from the other parent confirming travel is agreed and the birth cert.
I am currently in the process of changing back to my Maiden name but you need to change it by deedpoll if you want to change your name on your driving licence ad tax credits etc. at the moment i have 50% of my stuff in one name and 50%in another. Once my solicitor is properly open i will finish my deed poll application. It’s not expensive i just need her to authorise my witness forms are all online. My passpost says my maiden name and my driving licence is my married name. It’s getting very confusing.