Hello and welcome to Rootschat!
Both fostering and adoption were informal in the early years of the century, so there are unlikely to be any paper records to find.
A few things that occur to me:
When was your mother born in 1911? Does she appear on the 1911 census?
Have you the names of the foster family, and where are they on the 1911 census?
Sometimes unmarried mothers paid money for their children to be looked after, as they were unable to go out to work otherwise. I believe prospective foster parents sometimes advertised for foster children.
However, it may be that there is some family connection somewhere, and the only way to find that out is to research the foster family's history to rule that in or out.
Unfortunately, the 1921 census is not yet released, so you are unable to check where the foster family/your grandmother is at that time.
Otherwise electoral registers may give you some information, although of course your mother won't appear on them until she reached the age of 21, when she was married anyway.
This is the sort of (relatively) recent history that can be difficult to pin down, unless you have thought to ask the right questions of the right people at the right time, sadly.