She should not passively wait to be offered the flu jab as she may be waiting a long time! In my local area we can have the NHS flu jab at our GP, booking hospital or Boots, if you bring proof of your pregnancy e.g. scan report. I had mine at Boots at about 12 weeks.
As her employer you need to conduct a pregnancy risk assessment with her, establishing what she can and can't safely do. You can re do it later in pregnancy if her circumstances change e.g. back pain.
I am a HCP and at my risk assessment we agreed which types of patients it would be potentially unsafe for me to see. There are some illnesses that can be dangerous for the developing foetus e.g. chicken pox (if she does not have immunity from having had chicken pox herself). Also we agreed preventative measures e.g. not lifting heavy objects, regular walks around the office to aid circulation (rarely a problem as a Nanny I'm sure!).
You can probably find risk assessment templates online.
You do need to agree on what is reasonable though, and you probably need to have a frank discussion about this with your Nanny to reach a consensus that goes into your risk assessment. If your children were very unwell and likely had flu that is one thing, but if it is just a sniffle it is much more likely to just be a cold, and whilst a pregnant woman would feel lousy with a cold, it wouldn't be specifically dangerous to the foetus (unless she got very unwell for some reason).
Although it sounds like she is being unreasonable, emotionally I can understand her actions- I felt a fierce need to protect my unborn baby from even the tiniest risk, and I also felt so tired that I didn't want to risk even a cold. Luckily my job was pretty quiet and there were so many other staff that I could opt out of lots of things without causing problems, even though they weren't mandated in my risk assessment. Obviously a Nanny job isn't like that, you don't have ten other nannies pottering around the house, ready to leap in!