If your dm was sent away to give birth it's unlikely that the birth will have been registered in your local registry office but, even it was, you won't be able to view their registers and will need to make the necessary application and pay for a copy certificate.
The National Register Office holds the registers of all births/deaths/marriages that occur in England/Wales. The registers of births are divided into quarters - Jan-Mar, Apr-Jun, Jul-Sept, Oct-Dec - and it's important to note that, for example, the name of a child born in say, mid to late February may not appear in the expected quarter's register but will be found in the register for the following quarter.
Similarly, the birth of a child in, say, December 1973 may not be be registered until January 1974.
Where a child is adopted, a copy of their original birth certificate can be obtained either from the registry office that registered the birth or from the National Register Office .
Adoptions are recorded separately in yearly registers; copy adoption certificates can be obtained from the NRO but it is necessary to know the new name of the adopted child to verify that they are one and the same as the child whose birth you have found in the general registers..
Pre-1980 the registers of adoptions are massive tomes but they became progressively slimmer and lighter in subsequent years.
Secrets and lies are commonly practised in families and it's entirely probable that a child who was adopted out of your family traced their birth mother, and met your dm and dgm who saw fit not to tell you or your younger sibling, Ycq. I sincerely hope that, on receipt of proof that you have a half-sibling, you will welcome them into your family.
I've helped many adoptees trace their birth families and, sadly, your experience is not uncommon, auntevil, but your subsequent contact with your birth mother can't be said to have caused her any shock other than that of your existence possibly being exposed to her h and other dc she has had since she gave birth to you.
As it's also not uncommon for more than one illegitimate child of the same mother to be given up for adoption, I recommend checking the registers of births for the years prior and subsequent to the birth of a child who was given up for adoption until such time as evidence of her marriage and the possible birth(s) of her legitimate dc can be found. .
It's important to bear in mind that pre-1975 it wasn't uncommon for twins/triplets etc to be separated at a young age and adopted by different families.
Should any mumsnetter be embarking on the search for their birth family, or be in need of any assistance genealogy-wise, please feel free to pm me.