There is no national carer register. There is no official carer card to carry around. You're conflating so many things. There might be carer /charity organisations that have mocked up their own card, google and try them.
But if there was a national register for informal carers one, fed by GP, DWP or Adult services/Local Authority information , then based on info you've provided so far- you're aren't likely to qualify as a carer. Your mum would need to require 'substantial and regular care' help from you to meet LA and DWP criteria and she doesn't. A bit of shopping and laundry at the level you indicate for reasons you indicate , isn't that.
It is sad and very lonely times during lockdown. And lots of people have been advised to stay home for longer than lockdown periods and your DM might be in a vulnerable groups.
But Given what you've said, it's unlikely you'd be eligible under carers assessment using the Care Act as you fall at the first criteria since your mum doesn't have Care and Support needs as defined under the Care Act (which is quite a high level)
You won't meet carers allowance DWP criteria for carers allowance (even if you don't claim it) if she's not eligible for AA attendance allowance or PIP.
GP surgeries want to know when someone is an informal carer (they used words they'll 'register as a carer' on their internal GP records) as it can impact on their own health and as important contact details for medical records. It's not apparent DM's health level would trigger that though & you'd get a strange look if you ask and she usually presents as perfectly mobile, capable and with capacity.
OP you've admitted you're looking for a reason to visit your DM, that her NDN have offered to do her shopping, cleaning and laundry for her. Ask yourself why you think it's safe to go on i2 tubes across town to do laundry and shopping that a neighbour could? As that would expose her to a lot more risk of virus and Your mum has a heart condition. Most people are dropping shopping (& could do same for laundry service) on relatives doorstep just so they avoid increasing risk to them .
Ultimately You don't need to prove essential care to an vulnerable over-70-relative but police might rightly ask why you have to go on the tube and travel so far when a local organisation /neighbours could do those things for her and that's for you to ask yourself,