A child being made legitimate by the later marriage of the parents was introduced by the Legitimacy Act 1926. At that time it was optional to re-register the birth, but there was a condition that both the parents had to have been free to marry at the time of the child's birth (this was to prevent the law being seen to condone adultery apparently). The birth registers of 1927 and 1928 are full of re-registrations under this act, with some of the births having taken place many years before. I have an example of a child born in 1892 whose parents married in 1894 and who eventually got around to re-registering the birth in 1939.
The couple have to be the parents of the child - a re-registration can't be used to legitimise a step-parent relationship, nor does it ever have anything to do with adoption.
Even if the birth wasn't re-registered the child would still be regarded as legitimate under the provisions of the act.
That "free to marry" condition was removed in the Legitimacy Act of 1959, but re-registration was still optional.
The 1976 Legitimacy Act (s9) made it a legal requirement that the children must be re-registered, this is why a registrar will remind a couple who already have children when they come to give notice to marry, and usually also again at the ceremony itself. However many couples forget, or just don't bother, so it doesn't always get done.
Apart from the stigma, illegitimacy still mattered in terms of inheritance rules right up to The Family Law Reform Act in 1987.
The way the information on a birth certificate is shown makes it immediately clear whether the child is that of a married couple or not so the new certificate issued from the re-registration clearly shows that the child is legitimate and born to a married couple. Whether the mother has changed her name or not is irrelevant - a maiden name will be shown, even if it is the same as the married name she is using ( using "formerly" to indicate a maiden name hasn't been done for many years - the maiden name now goes into a separate space on the entry).
The child's name and surname can also, in some circumstances, be changed at the time of re-registration, although subject to quite a few restrictions.
As many couples do now marry after having children, these re-registrations are very common and happen all the time.
