I was born in similar circumstances after WW2. I got my adoption papers in 1977 - which showed my father to be "Jack Robinson, Army Driver, Age about 30" - total ficticious nonsense.
I discovered my father's identity in 2018 by matching my DNA with his grandson - then obtaining father's army records, which confirmed that he was in the right place at the right time!
The daughter in this investigation is a DNA match of mine.
She knows her paternal grandfather's identity, but not that of her paternal grandmother. Her father did not know the identity of his mother, as he was brought up by a variety of people, who were distant relatives, at best, being before 1926, as mentioned.
Using our shared DNA matches, I think that I have worked out the identity of her paternal grandmother, who registered the birth of a son, which would agree with the later birth date (i.e. not what is on the Army attestation). There is some agreement in the names, but not total agreement. Obviously, I cannot be 100% certain.
The daughter does not accept this, but relies on some purely circumstantial evidence, which was among her father's papers. Her choice of paternal grandmother only works for the earlier date of birth, for which there are no BMD records to support it.
Thanks for your comments and I hope this makes it a little clearer.
Gordon