If a vaccine is given when a baby still has antibodies to the disease, the antibodies can stop the vaccine working. This is why routine childhood immunisations do not start until a baby is two months old, before the antibodies a baby gets from it's Mother have stopped working.
This is also why it is important for parents to stick to the immunisation schedule, as a delay can leave a baby unprotected. A delay can increase the chance of adverse reactions to some vaccines, such as pertussis (whooping cough).
For details of all NHS vaccinations, for babies, children (and adults), please visit https://www.nhs.uk/vaccinations/
Details of what is given and when can be found at https://www.nhs.uk/vaccinations/nhs-vaccinations-and-when-to-have-them/
Further information on vaccinations is available from Oxford University: https://vaccineknowledge.ox.ac.uk/