Firstly Lea and Fleabag, my best move so far in a similar situation was to get a support worker. I got one because I went to a memory clinic held at my doctor's surgery, but if they don't do one, then ask at a local Alzheimer's daycare centre if you can visit for a chat, and ask them. Perhaps someone who lives in your area and is on this forum knows how to access one. A support worker will help you apply for everything you are entitled to, it's a first step you need to make.
Ask their local council if your mum/grandma has to pay council tax, even if it doesn't seem like any big savings, you need to apply for these things along with attendance allowance.
Lea, it may seem to you that your doctor is not being on your side, well that's what it looked like to me, until I thought about it from an outsider's point of you. Don't take it too personally if your grandma has been putting in a complaint about you to him. My mum came up with some horrific stories about her friend in her early middle stages of Alzheimer's. claimed her friend had a baby out of marriage and had sold it so her husband wouldn't find out, (she actually had a miscarriage and could no longer have children because of it). Claimed I should be grateful because she and my father adopted me, (not true in case anyone is wondering about smoke and fire). My sister has been accused by her of neglecting her small children, ( her youngest is 35 years old, and has just taken a year off work to travel around the world) Sounds like doctor has either been hoodwinked or has concerns that you may not be aware of, speak to him. Wouldn't hurt to ask him if he would be willing to sign a form from the council to back you up that your grandma has severe mental problems when you get the form from them. Try and find out what his concerns for your grandma's health are, use your Power of Attorney of Health to get an appointment with him at a time your grandma is not with you.
Best way to get a needs assessment if you want one, (and why not, it really doesn't hurt, provided you know that they won't pay for anything) is to ask about something you know they will provide. Certainly in Lea's case, I would ask if they would come and give you a needs assessment because you would like to find out about a panic button because you are worried that your grandma might get lost whilst wandering, or fall over and be unable to get help because of her mental disability, and that you wonder if she should have an alarm on her front door in case she wonders off at night. Don't do this online, don't go through any other source except for the Social Services, if they tell you that you will have to pay for it, explain to them that you do have power of attorney and you feel that this should come out of your grandma's money if necessary. You don't have to sign up for any of this unless you feel it's a good idea, but it should persuade them to come and listen to you. Ask for a meeting at your grandma's house if at all possible, and don't ask for her permission, sounds like she's not going to be very compliant.
I don't know what it is, but certainly round here this seems to be something that Social Services are trying to push regardless of how suitable it is. If they ask about your Grandma's money, tell them you will look into it, but can they come and do a full needs regardless. It doesn't really matter if they want to do a financial assessment as well, you already know it will tell you that your grandma is self-funding and is not entitled to very much, but actually, it is a very painless procedure, so if they want to make a few quid for doing one that they already know the answer to, let them. Having said that she would be entitled to handlebars to get up and down stairs, or into the shower, possibly a seat if it's a walk in one, a rail around the toilet seat if she is a little unsteady getting on and off the toilet. She should also be entitled to equipment to help her get in and out of the bath. Anything that is small is usually a freebie which strangely is not means tested, and if they have nothing left in the kitty for these things they will tell you to apply for them from Welfare, but let them decide on that during your needs test.
£400 a week sounds very cheap for a care home. Round here the cheapest room is £850 a week, and I live in the North where apparently it is supposed to be so very cheap. Whilst my mum is self-funding, I have opted for a live-in carer, which may be slightly more expensive than a care home because of bills as well, but it's not my money, it's my mother's money. I did put her in respite care for 1 week she hated it, I had to go down every day to sort her out because they kept ringing me, so I opted for live in-home care for her. You may feel right now that your grandma has 3 to 4 years left before needing help, but I think anyone on this forum who has had this going on for longer, will be able to tell you that things can change quite dramatically over a very short period of time without much warning, and the more they need help the more expensive it becomes.
Now you would think that Social Services would be keen for you to save as much money as possible so that you don't run out of funds, but I actually found during my mum's needs test, their main goal seemed to consist of advising me on how to spend as much money as I could out of my mum's money so as to get down to the capital threshold as soon as possible.
I have been told by the needs assessor that once my mum is no longer self-funding that they will recommend that she goes into a care home, and since she doesn't own a house, I think the decision is irrelevant of that.
One of the things my support worker told me, was that if you don't think you are getting listened to by the person doing the means test, you are allowed to ring up the Local Authority and ask for a second opinion, so if there is a clash of personality, you don't actually have to put up with it.