Based on the information you have given me and the government guidance on this matter, it appears you can add your partners name on the Birth certificate when you arrive in the UK. Firstly, you will have to complete a form. Secondly, because you and your partner are not married you must complete a Statutory declaration of acknowledgement of parentage form. Then register the change and if you want another birth certificate showing the new changes, you need to order a new one. I will link the relevant government guidance on the process, please follow the instructions. Please note this guidance is not exhaustive on the matter.  https://www.gov.uk/adding-fathers-name-birth-certificate