Cosmos, I understand why you want to know more about your grandmothers adoption, but I dont believe there will be any documentation for you to find. Child adoption had no legal status in Britain until 1926, when the first Act was passed which regulated this in England and Wales. Until then, child adoption was an informal and generally secretive procedure.
I have looked to see if there was any apparent connection / acquaintance between Annie Bessies adoptive parents and her biological parents, and I found nothing.
I believe we have now narrowed the adoption time down to a 2 year window between 1909 and 1911.
To try to narrow it further, you would continue to look for the death of Thomas Brown. With all the information we have now, I am inclined to think he died before 1911, rather than the other option of him deserting his wife and family. Knowing his DOB, did not throw up any new leads for him still living after 1911, that I could see. Others may have more luck.
There is another possible death for Thomas to look at that I mentioned above (reply #69). If you cant find a death for him before 1911 then I am not sure I have any more ideas on tracing Thomas. Unfortunately with such a common name, it is extremely difficult.
The other thing you could try, is to find out when Annie Ruth Brown started work at Woodcote House School in Windlesham, because her children would probably have been adopted before or around that date. The school still exists today and may have archives with records of staff employment that go back to the 1909-1911 period. They may not be able to help, as she was domestic staff rather than a teacher but you have nothing to lose, so asking them (and explaining your reason) is an option to consider.