Re-registration to add an unmarried father to an entry where he wasn't originally named is done under s10A of the Births & Deaths Registration Act 1953. It doesn't add to the original registration, but creates a new register entry, which can be months/years later.
It is an option, and not legally required, but a very common thing to do. It was introduced as a specific process in the 1953 Act, although you do see cases that were allowed previous to that by application to the Registrar General (who can authorise any re-registration, for any reason they deem appropriate).
Re-registering after the later marriage of the parents is done under s14 of the B& D Act 1953. Again it creates a new entry, it doesn't change the original one (other than a marginal note).
The Schedule to the 1926 Legitimacy Act said it was the "duty" of the parents to re-register the birth, but set no penalty for not doing so.
The current requirement for parents who marry to re-register is in s9(1) of the 1976 Legitimacy Act and states it should be done within 3 months of the marriage, s9(4) sets a penalty on conviction for not doing so. However the Act also makes clear that it is the marriage that makes the child legitimate, not the re-registration, so failure to re-register doesn't affect that. Many couples don't realise it is required or just don't bother to do it, but there is no mechanism to enforce the provision so no action is going to happen.
Although it is required within 3 months, it can be much later - the longest example I have is of a child born in 1892, whose parents married in 1894. The "child" was 47 years old when the birth was eventually re-registered by the father in 1939.
Re-registration after marriage ( or civil partnership) can also now be done to add a "second female parent" and not just the father.
It is also not uncommon now for a birth to be registered 3 times over a period - once with the just the mother named (unmarried), again to add the father's details (still unmarried) and again once they do marry.
Re-registrations can also be done for other reasons such as in some cases of surrogacy, or gender change, and are allowed by the relevant legislation.