Was Ridgeway your mum's maiden name? If so, there is no birth registered for a baby born to a mother called Ridgeway between 1955 and 1965 in the correct area, unless she registered the baby with a different surname. I had a baby at Doriscourt whilst unmarried and the birth is clearly shown on FreeBMD, so I know if the baby was born at Doriscourt it would have been registered there, as the registrar came to the nursing home to register all babies (not just the ones of unmarried mums).
In any case, even if you get the birth certificate it will only show that the baby was adopted and there is nothing you can do, apart from your mum registering her interest with the Adoption Contact Register
www.gov.uk/adoption-records/the-adoption-contact-register and hope that the child has also registered. Mums cannot find adopted children, only the adopted children can find their mums.
Unless, of course, you try a programme like Long Lost Family and see if they can help.
ps. I haven't got my daughter's birth certificate, but I could get it if I wanted to as I can see the reference on FreeBMD. I know my daughter got her birth certificate so she could trace me, which she did about 12 years ago.