May be one of these, or even a mix!
For a long time - I believe since the first Registration act (1836?) the child of a married woman has been assumed (not just socially - legally) to be her husband's child. She has to state that it is not, then see below.
An unmarried woman can only register the child as her own. The father of the baby must be present if he wishes to name the baby as his own (there are specific procedures for men in prison / serving overseas).
There are checking procedures between maternity and registration systems to check that babies are registered
Parents can name their babies as they wish, there is nothing to stop them giving different surnames.
In 1979, it was fairly common for registrars to hold sessions at hospitals, and mothers who were still in hospital could register the babies, and save themselves a trip into the office. However, registrars were not trained to pick up anomalies, such as confusion induced by infection or post -anaesthetic, and there were a few problems (even on one occasion, the wrong gender). The practice was dropped during the '80s (not certain of exact dates in different regions). After that, any mother who too ill to get into the office, or unable to send her husband, can ask for a home or hospital registration, but there are checks to be carried out. I have known it happen a few times.