Sorry, I think YABU.
And I am saying this is as someone who has had a number of nannies and au pairs over many years. And, yes, most of them gave the children cards and presents. Indeed, the nanny we had the longest (many years) still sends my younger ones cards. I feel really touched when the cards arrive - we all still miss her and I wish very much she was still our nanny, but she doesn’t live nearby anymore.
She doesn’t send my oldest a card, even though at one point she used to be his nanny too. That doesn’t bother me at all - I hadn’t ever dwelt on it until your post made me think. There is a big age gap between him and the others, and she knew them since they were babies. The relationship is different.
Last year we had a new nanny who had been with us a short time when one of the children had a birthday. She didn’t give a card but made a fuss over the child and asked questions about the party that was planned. I thought no more of it, neither did my child.
The thing is, there has never been an expectation on any of them to buy my children cards or presents. I would have been mortified if I thought there was.
All our nannies and au pairs were employed in a professional capacity and if my children were being looked after and cared for, which they were (mostly, but that’s another story!), then I’m happy. Over time, bonds and friendships can develop, but that is a wonderful bonus and not something I expect.
So, overall, I have to say that I do think it is unreasonable to be put out over a nanny not getting a child a birthday card. While it is lovely if it happens, I don’t think it is a reasonable expectation. Even more so when the nanny is still relatively new and this is not even the child she looks after most.
If everything else is going well, I wouldn’t fret. No, it is not a yardstick of her affection - even if her relationship with the children could be measured in that way (I don’t think it should be), she hardly knows you all after only four months.
You’ve many years of childcare to navigate yet and finding someone who works with your family is worth much more than a card that will scarcely be noticed.