I think Buffybee is being optimistic. You need to first check whether you will be allowed to turn the garage into a habitable room. There are circumstances where planning permission will not be given for such conversions.
It will be worth while getting pre-planning advice from your local council. This will cost about £150 and should include a site visit. They will discuss, not only whether you will be allowed to convert, but also any limitations and any extra work that might be necessary, such as upgrading insulation in other parts of the house, or extra building work needed to make it possible to adequately insulate the conversion to planning requirements - possibly needing to replace the garage roof.
Building work takes time and if your mother needs to sell her house before work can go ahead, however quickly you get planning consent, it could take six months or more to sell her house and where would she live while the extension was being built? Taking selling and building, it could be a year before she could move into the converted garage.
If your mothers life is, sadly, limited, it may be that her suggestion that she rents a sheltered house may be the best solution. She can move in while her house is selling. Or, if they can afford it, that she and her DH go into a care home together.
Long term, a garage conversion might have been a good idea, but I suspect you need an urgent solution to make the last few years of your parents relaxed and happy so rented sheltered property or even a care home for both may be the best solution.