If you are not named on the birth certificate you don't have parental responsibility, which means that the only way you will be able to insist on seeing the child will be if you a) get the mother to acknowledge that you are the child's father and jointly complete Form C(PRA1); b) get an order from the court declaring you to be the father (form C1), or c) getting a child arrangements order for the child to spend time with you, (form C100).
The choice as to which route you take is up to you.
C(PRA1) Form: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/961321/c-pra1-eng.pdf
You will need to attend a MIAM (mediation) before you can apply on a C1 or C100. Cost is £215 for the applications (each) but if you are on a low income/benefits, see the EX160A information sheet and application form for fee remission.
Find a mediator here: www.familymediationcouncil.org.uk
C100 form: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/946762/c100-eng.pdf
EX160 form: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/952045/ex160-eng.pdf
E160A Notes: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/958038/ex160a-eng.pdf