Birth registration and baptism are two entirely separate things. Your grandmother was baptised, not at St John of Wapping but at the church of St George in the East (on 1 October 1882). Her birth was registered in the St George in the East Registration district in the third quarter of the year. There are no other COOK baptisms at either church in the relevant time frame. Nor are there any birth registrations for a likely child.

The time frame is even narrower than you suggest, as there is no child with the couple in the 1881 census. It is theoretically possible for Eliza to have had both girls in very quick succession (though very rare among the working classes at the time, when breast-feeding - nature's contraceptive - was the only option), but such superfertility does not tally with the relatively long gap between marriage and the first child.

Carol's earlier suggestion of a stillbirth is a much more likely scenario and accounts very tidily for the lack of both birth and death registrations. Church burial registers will not help here, I am afraid, as the London churchyards were closed by Act of Parliament some three decades earlier. If Adelaide had been born alive, her death would have been registered and she would subsequently have been buried in one of the municipal or private cemeteries which had been built on the edge of the metropolis. A record should have been made of the burial, though it is very unlikely that there is any permanent memorial as such things were beyond the means of most of the urban working classes at the time.

This seems to be one of the cases - very common - where family lore does not fit with the documentary evidence. Might it be time to re-evaluate what you think you know?

Caroline