Well, the heresay about an Isle of Man connection may be right if this is the right Dora. Actually she is the only Dora Quale I found in Wales.
1901 DISTRICT 4, LEZAYRE, ANGLESEY, WALES ADDRESS - 35 NARRADALE QUAYLE
John J., age 46, farmer, born Lezayre, Isle of Man
Eliza J.C., wife, age 45, born Jurby, Isle of Man
John C., son, age 13
George S., son, age 12
Eva C., daughter, age 10
William, son, age 9
Robert C., son, age 7
Thomas H., son, age 4
Dora M.,daughter, age 2
all children are shown born Lezayre, Isle of Man
I tried to find Stanley Darwell in 1901 census but did not so he may have been born after that. Or he may have a different first name and perhaps put a middle name on birth record of your mother.
You mentioned your mother's birth cert. showing her born Wales, but I found a birth for a Constance J.M. Darwell Jan/Feb/March 1920 in Bangor, Greater London Mother's maiden name Quale
Here is the record from ancestry. This is from ENGLAND AND WALES BIRTH INDEX 1916-2005 ***********************************
England & Wales, Birth Index: 1916-2005 about Constance J M Darwell Name: Constance J M Darwell Mother's Maiden Surname: Quale Date of Registration: Jan Feb Mar 1920 Registration district: Bangor Registration county: Greater London Volume Number: 1b
This is very odd with you having a birth record showing her born Wales. I wonder what the copy of this record would show for father.
The London database of birth record images only goes to 1906, so I could not find the actual record with image.
There is a family tree on ancestry.com which shows a Dora Mylchreest Quayle born Lezayre, Isle of Man 1898 but this tree is personal which means it is not open to the public so all I could see was that bit of info - the rest of the tree is hidden. I tried searching for Dora's father John and he is in the same private tree. I have sent a message to the owner of the tree asking if they have any further info on Dora. I'll let you know when I hear from them.