Adoption and birth certificate in the UK circa 1938
I'm writing a story set in the UK about 1978/1979. I have an OC, born in 1938 and adopted at birth. It's my understanding that the UK was moving towards legal recognition of adoption with laws in the 1930s and 1940s. I know that there are short form birth certificates and long form birth certificates. Will the long form certificate indicate the person's natural parentage in the 1938 register, or might he be plausibly registered under the adoptive parents' name? My character is an employee of the Ministry of Defence in the 1970s, and for plot reasons I need for the MOD not to be aware of his birth parents. If the registry of births does normally show that this character is adopted then I need to change the adoption in my story from a legal adoption to something more under the table. The adoption being under the table would suit my plot. I just need to know so that I'm not talking through my hat about what people might have expected at the time.
Terms searched - adoption United Kingdom (with variations of 1938 and 1940s), history of adoption United Kingdom (similar date variations as previous), plus various combinations of birth, registration, does birth certificate show adoption, etc etc.
ETA: Thank you to the commenters - your input has helped me finalise how I want to approach this plot point in this story. Much appreciated.