You are clearly going through a very difficult time and could do with some help to deal with it. Losing your mum 5 days after losing your nan must have been devastating for you and will clearly have a big effect on you now as well as back 9 months ago. Because your mum died a little while ago, you might feel that people think you should cope now and get on with life and you may find it hard to show your feelings openly. There is no right amount of time for when the hardest sadness is over and the grief gets less overwhelming. People often think it takes longer than they expected and then they think something is wrong when the sadness is still there after a long time. You will never 'get over' loosing your mum, as it will stay with you in different ways for the rest of your life. But you will learn to live with your loss in a way that is not dominating your life.
As you describe, your whole situation is more difficult because your dad has got a new partner. This could have been a support for you, but it clearly is not. There seem to be many reasons for why this has such a negative effect on you. First of all, the fact that it happened so quickly after your mum died must have been very difficult. This would also have made it harder for you to share your feelings and grief with your dad, as his attention would also be directed towards his new girlfriend, and not just towards you and his own grief. As I don't know the relationship between your mum and dad, it is hard to know how he could move on so quickly. Some people do this in order to escape from their grief as an attempt to get on with life as quickly as possible. People can also be very vulnerable and emotional when they are grieving and may feel lifted and more positive by the affection and comfort that another person can offer.
It is really sad that all this has had such a bad influence on you and your relationship with your dad and also with his new partner. I think it is very good of you to acknowledge the situation and wanting to improve it for the future. You are clearly a very considerate girl who are thinking about how you can make your dad more happy too. You are absolutely right, it will benefit you all if you could manage to get on better. You say that you don't like your dad's new partner because she is selfish and sly. She might feel very strongly that you don't like her and this might have made her act very defensive and maybe negative towards you. I am sure she will have other sides to her as well and they will probably come out you can get more relaxed around each other.
As usual in these stuations, an open communication is often the best way forward. Tell her and your dad that all 3 of you need to sit down and talk about how things are between you. I would imagine, the reason he doesn't normally want to listen to you is probably because he knows that you will critizise his partner and he has a lojalty problem with that. Try to explain what has been difficult for you, without getting negative about her as a person and if possible without throwing too much blame on them. It is much easier to talk in a friendly way when people don't feel threatened and are made to feel guilty about things. Through a friendly chat, you might discover other sides of your dad's partner too. Also ask them how they think things can improve between you. They are the adults and should take their responsibility with this. It could also be a good idea to maybe do something with her alone, just the two of you so that you can get the chance to talk and maybe bond in a different way then if your dad is there. It would be good if you could manage to sort this out between you in the family. If this is not possible, there are local places you could seek advice for family counselling. You should find information about such services in the support section of the riprap website, in the section called 'support in my area'. There are also organisations that are working specifically with children and bereavement and you can find information about these in our links section.