Family problems who is in the wrong?

It sounds like your father is in the wrong and habitually unpleasant by nature.
His behaviour seems ultra critical, highly negative, coersive, controlling and manipulative.
Don't allow his bullying of your mother or gaslighting of the rest of the family to always pass without direct challenge.
Consider replying to his nobody wants him type of comments with a firm "that's right- nobody wants your insults or bullying. Go away, take all your vicious babble with you. Your nastiness won't be missed".

I agree with londonmyst, although my style would tend to be to try to be less combative towards the father.

My sugggestion is that you do what you can to give your mum as much genuine positive reinforcement as you can. Whenever you can give her a genuine compliment do so. Whenever you feel like saying that you love her, do so. And tell her why you love her as often as you can.

For your dad, next time he starts criticising her, step in and say "Dad, how about you saying something positive to Mum, to balance out all the criticism you've been giving her? She is, after all your wife and my mum."
Or totally short circuit him by saying "Dad, how about you and mum going for a week-end away? As soon as the hotels open up again? Just the 2 of you. To a hotel with a swimming pool. I'll look after things here, whilst you are away." Don't get into an argument with him about this. Just put it out there and let him respond however he wants. You never know, in 3 months time he might say "I've had this great idea. Me and mum will go for a week-end away..."

You may also have to be realistic. As parents tend to be really pig-headed when it comes to taking good advice from their children. And it may be a case of you making the best of a bad job.

Which would be to learn some important lessons from this.

Criticism is the poison drip-feed in a relationship. Eating it away, drip by drip, cutting comment by cutting comment.
Aim to minimise your criticism of the people you love. A limit of criticising them twice per year per topic of criticism is a good target to aim for.

And avoid relationships with people that are overly negative and critical towards others. What you'll often find is that they will be fine for the first 6 months of your relationship. And that the criticism will start coming in after that.
A good tip for seeing through this common facade is to visit their family with them. And watch how they talk and interact with the parents and any brothers or sisters. How they talk to them will become how they talk to you, once the honeymoon period has worn off.

I do say such things but then in the end I'm left feeling like I'm the nasty one, he even called me abusive once

I agree with londonmyst, although my style would tend to be to try to be less combative towards the father.

My sugggestion is that you do what you can to give your mum as much genuine positive reinforcement as you can. Whenever you can give her a genuine compliment do so. Whenever you feel like saying that you love her, do so. And tell her why you love her as often as you can.

For your dad, next time he starts criticising her, step in and say "Dad, how about you saying something positive to Mum, to balance out all the criticism you've been giving her? She is, after all your wife and my mum."
Or totally short circuit him by saying "Dad, how about you and mum going for a week-end away? As soon as the hotels open up again? Just the 2 of you. To a hotel with a swimming pool. I'll look after things here, whilst you are away." Don't get into an argument with him about this. Just put it out there and let him respond however he wants. You never know, in 3 months time he might say "I've had this great idea. Me and mum will go for a week-end away..."

You may also have to be realistic. As parents tend to be really pig-headed when it comes to taking good advice from their children. And it may be a case of you making the best of a bad job.

Which would be to learn some important lessons from this.

Criticism is the poison drip-feed in a relationship. Eating it away, drip by drip, cutting comment by cutting comment.
Aim to minimise your criticism of the people you love. A limit of criticising them twice per year per topic of criticism is a good target to aim for.

And avoid relationships with people that are overly negative and critical towards others. What you'll often find is that they will be fine for the first 6 months of your relationship. And that the criticism will start coming in after that.
A good tip for seeing through this common facade is to visit their family with them. And watch how they talk and interact with the parents and any brothers or sisters. How they talk to them will become how they talk to you, once the honeymoon period has worn off.

Don't allow any of his insults, demands and nasty lies to make you feel guilty or upset.
Remember that being a vicious tyrant and using emotional blackmail are your father's tools for making himself feel powerful.
Stay calm, remain in control of your thoughts and don't let him make you feel angry or exhausted.

While you have to share accomodation with him during the pandemic, develop coping strategies and do what you can to restrict contact opportunities to reduce the number of unpleasant interactions.
Focus on your health, maintaining a positive mentality, favourite hobbies and future.
Your ambitions, health, improving your finances, doing the things that make you happy and possible ways to achieve the future life that you want.
Good luck!

Thanks you make some very good points there. Thing is my mum doesn't want to spend that alone time with him when i suggests it because he's so difficult/unpleasant so she avoids it. Plus they are totally broke now so they'd never go on a trip anyway. My dad doesn't even leave the house ever. he has MH problems...but won't seek support

Your mum and you may have to take the lead on actually booking and paying for a week-end away. You can get very good deals on hotels with swimming pools on Sunday nights outside of school holidays in the UK. It's worth trying a short change of scene to see if that jogs any change in your father's disposition.

Although being realistic it does sound as if your parent's marriage is effectively over. All that's needed now is for one of them to break out from the dogmatic stubborness they've had up till now about not getting divorced.
A divorce would be, by far, the best course of action for your mum.

With your dad having mental health issues, it's only fair to cut him some slack and understanding.
Whilst at the same time being realistic for yourself and your mum. IE you should still love him, because he's your dad. But you should also be looking to get your mum and yourself living in a separate home to him because you love yourself and your mum.
If your mum refuses to leave him, then all you can do is go ahead and leave home ASAP.

I think it would also be worthwhile you discussing this with a trusted teacher at school / social services.
They may not intervene at this stage. But it's worth making them aware of the situation in case it gets even worse.

I think it's also important for you to try to maintain a logical / positive view of all this.
You going through this hardship now will help to develop you into the wonderful adult that you will become.
The quote about "That which does not kill you, makes you grow stronger" very much applies here.
But then again the principle of not just sitting there and being a victim also applies to you and your mum and your dad.
He should seek all the help via the NHS that he can get.
Your mum should divorce him.
You should leave ASAP to go to uni, or work in another city or even to be in your own 1 room council lodgings when you are at 6th form college.

Health and safety is the biggest priority in our adult lives. Without your health it's difficult to enjoy life.

Living with your father is adversely affecting your mother's health.

If she were to live alone, would she have the right outlook / mental toughness to maintain her physical and mental health?
If she would then there is no debate to be had as to whether she'd be better off if she were to divorce your dad.