The people responsible for this are Adult Services within the social services department of your county or unitary council. You probably knew this already. The way it seems to work is they try and reserve what limited funding they get from central government for priority cases. Don't know how priority is established. Grants are means-tested so they will try and encourage individuals to use their own resources, pointing out it can be done quicker that way (true).
The charity sector is also active in this area, providing some sources of funding locally. The social services people are likely to know which charities are possible sources in your area.
If I was looking for independent advice on how to maximise my chances, I would get in contact with the charity sector for advice - try starting with Age Concern (who surely most know something about it from an age standpoint) or try one of the national disability charities (say Shaw Trust) - not because you are after a grant, but say you are seeking advice.