Attendance allowance has no Mobility component.
The main beenfit for you to claim would be Carer Allowance. Your mum already receives what is called the "qualifying benefit" that classes her as "having a severe disability" and is th emain hurdle to Carer Allowance, since to claim CA the person you care for must get one of these "qualifying benefits".
However, be warned that Carer Allowance is a pittance - it is £61.35 a week. The only means test is income from employment - you can't earn more than £100 per week (after certain allowance are made) from working but other income (such as interes ton savings, dividends from shares, etc) is ignored, as are savings. It is also taxable income. Carers as carers get little else and there is certainly no Mobility allowance or car scheme for carers.
Depending on circumstances it might be possible to get help with council tax, housing costs such as rents, etc.
But AA and CA are the main help available and they are hardly generous.
Also be aware that your receiving CA may reduce your mother's benefits - it will not change her state pension, private pensions or AA but could affect others (for example, disability premiums attached to Pension Credit and so on)
Being honest I would think long and hard about this. Your mother may cope very badly with moving to a strange place. You also say she is hospitalised because of delusions. Some people have good outcomes and the drugs used arte effective...but not always. I would be personally inclined to wait until her prospective discharge and to see what a "best interests" meeting would suggest (you can be included in that). Is your mother hospitalised under section? If so what section?
If she is under section 3 and is discharged she would be entitled to 117 funding, in other words the NHS would entirely fund it including placmen tin a care or residential home.
My worry is that you will burn your bridges by leaving your job, your mother comes to live with you, then you find you cannot cope either right away or if she gets worse.