It's just one of those curious things that we come across during our research.
Indeed it is, and I've continued to wonder how baby Doris arrived with the Dyer-Foster family, assuming that the child in the 1911 Census is the daughter of Ethel Daisy Gregory (which seems strongly the case). In the latest newsletter Peter observed that Ann Dyer nee Foster had two older children born in Edmonton and pointed out that this is not far from Walthamstow. But I believe there is a much closer geographical connection.
Amy Hilda Dyer aged 8 in 1911, was born in the Walthamstow registration district and baptised on 03-Sep-1902 at St Mary the Virgin Walthamstow. As Peter noted, this suggests that the Dyers had lived north of the river, but it could also mean that they had relatives in Walthamstow with whom Ann stayed temporarily during her confinement. Sure enough, they did have family there. Going back to the 1871 Census I found Ann Foster aged 1 with her parents Jonathan and Phoebe in Hackney. She had two sibs, Jonathan 9 and Elizabeth 7. I then found the marriage of Elizabeth Foster (father Jonathan) to Charles Henry Brown on 05-Apr-1885 at St John Walthamstow. Subsequent censuses show that they settled in Walthamstow and in both 1901 and 1911 this Brown family was residing at 151 Chelmsford Rd. It would be interesting to know if Amy Hilda was born at that address.
Meanwhile in 1901 the Gregory family were at 66 Beulah Rd and in 1911 they were at 692 Lea Bridge Rd. All these locations are within a kilometre of each other, as can be seen on an annotated map that I've shared at
https://www.google.com/maps/d/edit?mid=13hFuuRoD9eCmemgjuVE-iZzAI3mJjSkL&usp=sharing.
This makes it entirely plausible that the Gregory family knew the family of Ann Dyer's sister Elizabeth Brown. Connection via church, school or employment are all possibilities. And that connection could have led to baby Doris's relocation, perhaps with a financial arrangement, saving embarrassment for the Gregorys. If there was a financial arrangement, that may have been something that the Dyers wanted to keep quiet, hence Doris was recorded as their biological child on the census.
On the other hand, there's Peter's innuendo that John Ernest Dyer could have been Doris's father. He was an artesian well engineer and a quick search shows that there were/are artesian wells in Walthamstow. Perhaps he was working there on site. As an entirely left field thought, could he have had the nickname "Wells"? He died in Essex, but he was not the John Ernest Dyer who died in 1957 as claimed on several Ancestry trees. Other records show that that man was an accountant. Rather, his death was in 1925 and this is in keeping with Ann Dyer being recorded as a widow at her remarriage in 1930.
Unfortunately none of this gets Peter any further regarding the possibility of his grandfather having fathered baby Doris. Nevertheless, it has been an interesting research exercise.