Thinking about this... the written references the agency is referring to may be ones which the agency has obtained by contacting past employers directly... so those references may not have been seen by the nanny themselves.
I would have expected that type of reference to be verbal, rather than written - a nanny agency would often call past employers and go through a set series of questions. I would have expected the agency to make the respondent of such a phone call aware that the information would be passed on to their client (ie. you), plus may be stored for future use (and also passed to future clients).
Could the agency paraphrase to you a highlight of the contents of the written references? Perhaps that would not breach Data Protection Act.
Could you ask the nanny candidate to provide you with written references directly? As a nanny I have written references from families for whom I have worked, which I am able to send to prospective employers.
Looking through some agency Terms & Conditions, it appears that some of them say: "Verification of all information and references shall be the responsibility of the Employer." - Or other such wording.
So if that is a case, once you have been introduced to the candidate, you can ask the candidate to supply you directly with references.
Association of Nanny Agencies Code of Standards - alas not of much help, not all agencies are members of the Association plus I can't find reference to what access a client has to the written references obtained by the agency. Perhaps worth contacting them for their view.
Have found a document on the ICO website with regard to employment references but alas it's as per the Tinies ruling, in that it's about Subject Access, that is access by the person whom the reference is about, rather than passing a reference between a recruitment agency and the potential employer.
Hmm, this looks as though it all may get a bit complex. Maybe you could ask the ICO... if the Data Protection Act is indeed preventing employers of nannies seeing references obtained by an agency working on behalf of that employer, then is that how the legislation should be applied... or is there some possible exemption?