Well generally the most basic relationship is nanny-employer. They have a contract between the two of them and that's that. Nanny also has a realationship with their insurer and possibly OFSTED.
An introduction agency brings nanny and parent together. They may advise on OFSTED registration and the like and mediate if things go wrong but they can equally wash their hands of the entire situation, having been a very expensive screening service.
An employment agency is an intermediary for the parent and nanny. So parent pays agency for x hours, agency finds, sends and pays nanny. Nanny is still registered separately with OFSTED but may either get their insurance separately (this is definitely the case with 2 employment agencies) or via the agency's umbrella policy (and there's definitely one that does this). In the latter case I'd expect the contract you have with the agency, and the one agency has with the nanny, covers medication and permission. As to the former, I don't know how that would work.
In all cases where the agency is the employer and the parent is the client, the agency would be able to redeploy the nanny without a need for references. They would, as the employer, be the one giving out a new reference. One such agency would also be able to shift the nanny out of nannying into another setting operated under the same umbrella. One hopes they'd deal with any complaint responsibly.
The relationship on whose say goes is more complex. I suspect in that case it would operate more of a nursery style scenario where nanny is assigned a line manager and if they have concerns they would talk it over with them, so a reasonable instruction would be followed but something like giving calpol when not indicated the nanny would ask the agency and follow their instructions, and you as the agency's client would take up their employee's conduct with them. If that makes sense. Just as a nursery nurse would defer to a room leader who might consult the manager if in doubt, and if you had a problem you'd take it up with the room leader or the manager.
If you're not clear on who is working for who, and I'm guessing you're the parent in this scenario, it might be worth asking for some more detail about the setup and checking the terms you have with the agency.
These kind of agencies provide a very interesting perspective on childminder agencies and how they might develop, and to what extend the relationship between CM and parents could be affected by a third party. It may be that more nanny agencies shift from an introduction role to an employment service, but I think there will always be room for parents to employ a nanny privately and it's likely to remain the more popular option for the foreseeable future.