There are two parts to this question
- Registration of birth
- Parents (or others) obtaining a birth certificate (copy)
There are two distinct documents that are colloquially referred to as a birth certificate:
- "Certificate of Birth" (short-form birth certificate, abbreviated certificate)
- "Certified Copy of an Entry on the Register" (long-form birth certificate / full certificate)
According to Wikipedia:
In England and Wales, birth registration with the state began on 1 July 1837. The birth was registered in the birth district and at the end of each quarter, the registrar sent a copy of all entries to the Registrar General. However, registration did not become compulsory until 1875.
So registration has been compulsory since 1875.
In the case of every child born alive after the commencement of this Act, it shall be the duty of the father and mother of the child, and in default of the father and mother, of the occupier of the house in which to his knowledge the child is born, and of each person present at the birth, and of the person having charge of the child, to give to the registrar, within forty-two days next after such birth, information of the particulars required to be registered concerning such birth, and in the presence of the registrar to sign the register.
I can find no officially sanctioned circumstances in which registration within 42 days can be avoided legally.
The registrar estimates there may be 20 births a year that are not registered within 12 months.
Any person wanting a full form birth certificate nowadays has to purchase one (currently £11). Typically parents purchase one for their child. However it isn't currently compulsory for anyone to buy one.
At the time of the 1874 act, the maximum price was one shilling:
Every minister or person who performs the rite of baptism shall deliver the certificate required by this section on demand, on payment of a fee not exceeding one shilling.
Note "on demand", it wasn't issued automatically.
I believe the short-form certificate (omitting details of parents etc) is normally issued free to the parents. Obviously this can be accidentally left behind, lost or mislaid.
According to the Guardian
If you were born before 1983 then it is most likely that you’ve only ever seen your short-form birth certificate (the one printed in red ink). That’s because until 1983, the short-form certificate was considered evidence enough of a person’s name and date of birth. The short-form certificate is actually titled the “certificate of birth” and contains a person’s full name, sex, date of birth and registration district of birth. It is given free of charge when you register a birth. What we call the full birth certificate is actually a “certified copy of an entry on the register” and is the certificate that contains every bit of data you might actually want protected.
So the usefulness of the short form birth certificate changed at roughly the time you ask about. However there is no reason why a birth in the UK would not have been registered - this would have been very unusual, and no special reason why your parents would not then have been issued with a short form birth certificate for their newly born child.