This is written from the employer's perspective but it basically confirms what I thought the answer to your question would be, in that it shows that such individuals are not subject to any form of registration and/or regulation:
https://www.which.co.uk/reviews/later-life-care/article/home-care-guides/how-to-employ-private-carers-alcB07s4rsFQ
There are very few occupations where there is a legal requirement for the worker to take out insurance cover (although, obviously, it would be wise to consider it) and private care work is not one of them.