Kids do get sick - especially babies, particularly at the most inconvenient times...
So, whatever agreement you have with the other mum needs to take that into account. As it's your nanny, who you have taken on the long term commitment of employing then it's reasonable that you get 'first dibs' on her whenever there is a problem such as this.
If your friend had her dc at nursery and they were ill, she would either have to take the hit and work from home or find emergency childcare.
You would be completely reasonable to tell your friend that, should this situation happen again (because let's face it, it will), then she is going to have to take the hit and sort something else out. At the moment you are doing her a massive favour which she seems to have conveniently ignored forgotten - she is able to have occasional childcare without any regular commitment to pay for a place. If she wants to be able to rely upon her child being looked after when the child is well, then she needs to fork out for regular childcare - be it a nursery, child minder or nanny of her own. If she doesn't want to fork out for it, then she has to accept that, with the best will in the world, if somebody else is paying for the nanny permanently (well, you are currently) then they get to have that nanny to look after their child exclusively when they are ill. She has work she has to do - well yes, so do you, so she can't trump you on that score.
Maybe if she was say going to a funeral so she was only going to be gone for a couple of hours and would come straight back to pick her child up that would be a bit different but just for work - even very important / busy day at work, then yes, you still get the nanny. If her other child was at nursery, does this child ever go to that nursery so could maybe go there as an extra day in an emergency? (I know that's dependent on their spaces / staffing though)
Take this opportunity to talk to her - nicely but explaining how this situation caused you lots of problems so that - together - you need to decide what to do about the situation.
It might be that if there is something that is really important to the other mum that she organises different childcare or has some back up emergency child care in place in case either child is sick or she realises that she has to take the hit or get her dh to or whatever. However, if she tries to assert that she should get the childcare regardless of the impact this has on you, your nanny and your dc then it really is a wake up call to the way that she think of you and how she values herself over you, despite you being the one that is helping her.