You need to know childcare law well.
We would offer emergency care in your own home or ours at very short notice
Care in the child's home, in England, is unregulated. If a childcare provider provides care in their [the providers] home, then that is regulated. It would be Childminding.
Why do the companies currently offering emergency care usually offer a childminder? Well my guess is that it is due to cost and due to the childminder having a space to take a child at short notice.
As a nanny, I simply do not have the capacity to care for a child at short notice. I am committed under contract to my employer. On my days off, then I might do it but it has to be worth my while as I am sacrificing my day off to a care for a child I don't know, at a location I don't know.
I have done nursery temping in the past, that is similar to what you describe. The nursery calls the temp agency at late notice to say they need someone fast - right now, or next morning. The agency then calls people on their list to see who can do it, get there for the time the client needs.
With nursery temping you get to know the nurseries after a while as you do repeat visits, if they like you they will request you.
Problem though is that for me as the childcarer, I had to be available very early in the day and often there would be no demand. So I would have no work but would have been ready (fed, dressed) since 6am waiting for the agency to call. The pay level (think it was £8 gross an hour back then, this is some 10 years ago) was good for nursery work but you didn't know how long you would be doing. Some times it may be 8 hours, other times it may just be for 3 or 4 hours. Very unpredictable, so was useless from the point of view of the provider in terms of being a steady income. I was employed by the agency, the agency sorted out Income Tax, NI, payslips and I had timesheets that had to be signed and returned to the agency.
You need to make it viable for everyone involved. That will mean it will be costly for the client. Would they pay enough to cover the cost wanted by the childcarer, to cover the admin costs (and taxes) of running the business, cover the advertising costs involved and resulting in a profit for you, the business owner.
The cost of 1:1 care is high. Thus why I expect many emergency care providers use childminders, as a childminder would be caring for a number of children and offers a lower per-child cost, then it would be to have a nanny on a 1:1 basis.