Can we back track a bit... from how I read it, you left the job... you didn't wait to be made redundant. Is that right?
If that was the case, then the parent could well be telling future employers that you left her employment for various reasons. While in an ideal world the written reference would reflect that but probably concentrated on the positives.
Have you been able to establish some exact things that she is telling people? Are those things just a personality mismatch, or did they effect your work?
Is she able to substantiate those comments... if not, then legal action may be possible, but may not be worthwhile.
As the job was longer than a couple of weeks, don't miss it from your CV. However you could only provide their contact details upon request and only in combination with the written reference and talking with your potential employer with regard to what you know may come up in a verbal discussion.
I would hope that future employers would take into consideration that an ex-employer may be disappointed that a nanny has decided to leave their job - thus that ex-employer may be holding a grudge and say some negative things about their former nanny.
What feedback have you had from agencies regarding this? They will generally be more experienced at reference checking and know when to take things an ex-employer says with a large pinch of salt, especially in situations where the nanny has left the job rather than being made redundant.