Your mother can obviously apply on Form UKM.
The fees are £80 (which is the standard fees for attending a citizenship ceremony) and she will become a British citizen after attending the citizenship ceremony (which will likely be held at a British diplomatic post close to her).
It is unclear whether you can register on Form UKM. As a general rule, British citizenship can only be inherited one generation outside the UK.
The
Romein judgment does permit double descent in case of Form UKM in some cases. However, the general consensus seems to be that it is only in the case of maternal grandfathers and not of maternal grandmothers.
In the case of the former (maternal grandfather), the mother would have been a British citizen by descent at birth already and therefore the question merely turned on whether she could have passed descent onto the next generation outside the UK by registering the birth with the British diplomatic posts closest to her.
In a case involving a maternal grandmother, the mother would obviously not have been a Citizen of the United Kingdom and Colonies (CUKC) at birth and hence would not be able to pass it on to their child (the mother) anyway.
Most of the online research that I have seen seem to suggest that you would only be able to register on Form UKM if the ancestor you weer claiming from was a British maternal grandfather (through a British citizen by descent mother).
A lot may turn on when your mother was born as the nationality laws changed significantly on
1st January 1949 as well as on 1st January 1983.
You could of course make an application and then proceed to take the refusal through the courts to the UK Supreme Court, as Ms Romein (herself also an American citizen) did in 2016.
As an aside, keep in mind that any British citizenship gained through Form UKM is prospective from the date of the citizenship ceremony and not retrospective.
EDIT: Some further quick research found
this link on this this very site, which in turn linked to a 2019
letter from the Minister of State for Immigration to the Chair of the Home Affairs Committee of the UK House of Commons.
The first question on Page 3 of the letter and its response by the Minister suggests that the Home Office is interpreting the Romein judgment and Section 4C to extend the ability to apply for British citizenship on Form UKM only to grandchildren of British maternal grandfathers and not in an infinite line of maternal descent.
I would also CC
@sunburn into this conversation. I believe that they have applied on Form UKM In the past themselves and they may be best placed to advise further.
Keep in mind that if you are intent on moving to the UK, you can apply for an Ancestry visa, which requires having one UK-born grandparent. After residing in the UK for five year, you can then apply for ILR, followed by naturalising as a British citizen in your own right a year later. This is a much better alternative if your have a family that you wish to bring to the UK.