Only those deaths certs after 1969 show a date of birth (if the informant knew it) any before that date would have an age on it.
"If the informant knew it".. this is where the problem might arise. When I registered my mother's death several years ago, I very nearly got the dob wrong - right day and month but wrong year. How many informants not so closely related have got it wrong, but believed they were right.
And if the registration in 1939 was informant driven (rather than the individual having to produce their birth certificate), how many dobs might be recorded incorrectly, either deliberately or through ignorance of the correct date.
My grandmother celebrated her birthday on the incorrect day (3 days different from that on her birth certificate) for as long as my father and his siblings could remember. It was a surprise to her when in her nineties and she needed to get a copy of her birth cert, that she had been wrong all her life. So she certainly would accidentally have given incorrect information in 1939. As she was born over a hundred years ago her record would be open, but there must be many younger people in the same situation.