I don't think the CT exemption is at all relevant, the conditions for getting carer's allowance are:
The person you care for
The person you care for must already get one of these benefits:
Personal Independence Payment - daily living component
Disability Living Allowance - the middle or highest care rate
Attendance Allowance
Constant Attendance Allowance at or above the normal maximum rate with an Industrial Injuries Disablement Benefit
Constant Attendance Allowance at the basic (full day) rate with a War Disablement Pension
Armed Forces Independence Payment
and you do the following:
You might be able to get Carer’s Allowance if all of the following apply:
you’re 16 or over
you spend at least 35 hours a week caring for someone
you’ve been in England, Scotland or Wales for at least 2 of the last 3 years (this doesn’t apply if you’re a refugee or have humanitarian protection status)
you normally live in England, Scotland or Wales, or you live abroad as a member of the armed forces
you’re not in full-time education
you’re not studying for 21 hours a week or more
you earn no more than £116 a week after tax and some expenses - these will be assessed when you apply
you’re not subject to immigration control
It's on the link below where you can apply on-line, as I say the CT has no bearing on the situation, she gets AA which is a qualifying benefit so as long as you do the above then apply.
https://www.gov.uk/carers-allowance/overview