Worryingly there is also this birth
Apr/May/Jun 1921
Basford 7b 577
Joseph A Lowe
mmn Hartshorn
So - either there is another Lowe/Hartshorn(e) couple around , which casts doubt on whether the Mary I found is the missing mystery daughter you were seeking
OR Milly has a child by someone else after John (Jack) dies, registering it with her current surname (Lowe) and her maiden name (Hartshorn(e)) but obviously Jack not the father.
Think you will need Mary's birth cert to see if she was the daughter of this marriage.