It is clear that you've done a lot of research on this topic. It's vast and complicated.
GRO should at least have the birth and death certificates for the colonies before their independence. I would also imagine they would have the registration documents as well from the 1950s. Alternatively, your mom can contact the DWP to see where they got her information from - or write to the Freedom of Information officer for DWP. If you can find registration, that's heaps easier.
Right of abode is the right to live permanently in the UK without any immigration restrictions. So, as long as the benefit is for a UK resident (and not for a UK citizen), I would imagine she would be entitled to benefits.
Your grandmother would have been a British subject by virtue of being born in the United Kingdom (s.1(1)(a) British Nationality and Status of Aliens Act 1914). Your grandfather also would have been a British subject (s. 2(1) British Nationality and Status of Aliens Act 1943).
Things start getting slightly trickier in 1948 because of the introduction of citizenship of certain Commonwealth countries and the British Nationality Act 1948. However, I still think both of your grandparents could be British citizens. If section 12 of the 1948 can be read as gender neutral, your grandmother would have been a UK citizen under section 12(1). Your grandfather would have also satisfied section 12(2) and would have also been a CUKC as he was a British subject at the time of the Act and his father was born in the United Kingdom.
Having thought about it a bit more, I reckon that as an alternative to a determination, your mother may be able to register as a British Citizen under section 4C BNA 1981 as a few years ago they got rid "born after 1961" bit. It's 80 quid and I reckon she might qualify:
Looking at Chapter 7...
http://www.ukba.homeoffice.gov.uk/sitec ... iew=Binary
Persons are entitled to registration under s.4C of the British Nationality Act 1981 if
(1) they were born before 1 January 1983 (your mother was born in 1950 and legally adopted in 1953); and
(2A)at some time before 1 January 1983 have become citizens of the United Kingdom and Colonies under one of the following provisions, if that section had provided for citizenship by descent from a mother in the same terms as it provided for citizenship by descent from a father...section 5 (see section 5(1) BNA 1948 - your grandmother was a British subject by birth in the United Kingdom and under section 12(1) BNA 1948 a citizen) or
(2B) at some time before 1 January 1983 have become citizens of the United Kingdom and Colonies under one of the following provisions, if that section had provided for citizenship by descent from a mother in the same terms as it provided for citizenship by descent from a father, and women had been able to transmit British subject status in the same way as men before 1/1/1949...12(5), of the British Nationality Act 1948. (
A woman who was a British subject immediately before the date of the commencement of this Act and has before that date been married to a person who becomes, or would but for his death have become, a citizen of the United Kingdom and Colonies by virtue of any of the foregoing provisions of this section shall on that date herself become such a citizen. Your grandmother was a British subject who at the commencement of the Act was married to a person who becomes a citizen of the UK and colonies by virtue of section 12(2)). and
(3) had they become citizens of the United Kingdom and Colonies as described above, they would, immediately before 1 January 1983, have had the right of abode in the United Kingdom by virtue of s.2 of the Immigration Act 1971 as then in force (see section 2(1)(b)(ii) - she would have been a citizen of the UKC legally adopted by a parent who had citizenship at the time of adoption and your grandmother had been born to a parent who at the time of that birth would have had citizenship by being born in the UK).
(4) I assume your mother would meet the good character requirement.
If you have the relevant birth certificates (mother, grandmother, great grandmother), adoption certificate and marriage certificates then this might also be a good route. If she's automatically acquired citizenship based on the documents she provides with the application, they return the fee as well according to Chapter 7.