I understand it's upsetting (and particularly now that she has to work through the notice period) but it's not your fault and it won't have harmed your son, so I would just accept it and don't stew over it. If she's leaving for that reason, it's probably best that she did as soon as possible anyway. Also you don't know whether the nanny made up, or overemphasised, that reason, because she had other reasons she wanted to keep to herself (like finding another job). Agree that 15m is a very difficult age for separation anxiety.
Maybe she's had a better offer and doesn't want to tell you.
Is she very experienced? Doesn't sound like it. It's not up to a 15 month old baby to bond with his carer, it's the carer's responsibility. Some babies, especially this age can take time to adapt to a new carer which doesn't mean he's unsociable. Children need a consistent and loving approach with lots of lovely activities to do together to help the bonding process. Is her approach compatible with your parenting?
I don't think she should have put it to you like this. Fair enough if she was struggling and had explained it in better terms. Has she mentioned anything about it before or asked for support?
Posted at the same time and just read your posts. I think you would have known if he had been completely miserable with his nanny. There probably would have been big changes in behaviour and a lot of distress when you left. Reading more I do think it's because something better came up.