Just one point: I only ever do the first round in a neutral location. I tried doing first rounds at our home when we recruited our first nanny and it was a bad decision - it seemed to create the wrong expectations on the part of the candidate. Also, I didn't appreciate at that point how important it was for ME to click with the carer as much as DD, and it's more difficult to assess this when you don't have five minutes to speak to them w/o interruption. I don't like telephone interviews because you have no sense of the chemistry IMO, but some people use this as a first step as well. So, I wouldn't be too freaked if part of the interview process doesn't involve the children.
Successful candidates - friendly, open, thoughtful and mature. Our current nanny, who knew we were interviewing a couple of other people at second round, said: "Look, you have to do what's best for your daughter, if you don't think the personality fit is right, I understand." I was impressed by that.
Don't kick off the interview with what does the job pay, I want X. Of course mention it though, if the parents neglect to, but want to bring you round again to meet the children. Our current nanny also asked some nice questions at interview, which made me think she'd really given some consideration to our needs.
Have names of references ready and be open about it - this includes telephone numbers of people. I wouldn't even bother producing letters - parents will want to speak to someone (should, I say) personally. I would never expect someone to turn up in a suit but professionalism that extends to references and original certs is really important. A copy of your CV handy is also good.
At the second round when they meet DD, it often really comes down to how you seem to connect to the child - everyone wants a scenario like Nick described - although it is possible to tell when someone is being really artificial about this. There's just no substitute for chemistry - sometimes either it's tehre or it's not. It sounds like a cliche but you've got to just be yourself. If you have a good feeling about the family, it will rub off.