OP you don’t sound like a harsh unreasonable taskmaster, but I would suggest that if she’s definitely very actively looking rather than just idly wondering what else might be out there for the future, and saying she is available without notice that she has clearly decided at this stage that the job with you is not for her for whatever reason to the point that she would be prepared to leave without a good reference.
So in your shoes (assuming it’s not obvious to anyone with half a brain cell why she wants to leave as you’re such an unreasonable dragon!! Which doesn’t sound the case! But in that case this advice will not work...) I don’t think you have much to lose by broaching the issue. I would sit down with her with a glass of wine and say in a non confrontational manner that you’ve become aware she is looking elsewhere which you understand she is totally within her rights to do at the end of the day. However as she had said she wanted a long term opportunity and you and the kids are really happy with how things are working out from your end, and you value her as an employee, you’re a bit confused that she seems to be thinking of leaving you so soon, and worried that she is unhappy, that you care about her wellbeing as she’s looking after your children and doing a really important thing as far as you’re concerned. That you wonder if she’s unhappy about some aspect of her job that could perhaps be adjusted to suit her better, if you know what’s going on. Maybe you agreeing to pay for the odd outing for her and the kids or the odd afternoon off now and again when you can manage it around work or something is all she’s really after and something you could reasonable easily give her. Nannies talk a lot to each other and small differences in what employers do for then can make a big difference to them I think.
You did say you would have expected her to raise any concerns or problems, but I’m not sure you are totally realising the power dynamic that is there in a relationship where you are basically her boss. I wouldn’t go running to my boss with every little thing I’m unhappy about in my job, particularly things that it is probably not in his power to “fix” (such as the salary he is able and willing to offer me for example, or things that fall well within the official job remit for this job which I’ve now realised I hate and there are other roles elsewhere where I don’t have to do that particular thing) or things that he doesn’t see as a problem (and/or in fact benefit him hugely at my expense) but are for me. Those are difficult conversations to have especially if you’re not a naturally confident person or are particularly economically (or otherwise like for a visa etc) vulnerable and at the mercy of your employer to some degree. You are not equals in terms of the power you hold (although nannies obviously do hold quite a bit of power compared to other employer employee relationships as having decent childcare in place is pretty crucial!), so it’s not just as simple as you’ll tell her if you are unhappy with certain things and she should tell you no matter how much you might want that to be the case.
Anyway just my two cents. If you’re happy with her I think it’s well worth finding out what’s not working for her to see if you can come to a new arrangement that she’d be happier with. I’m not sure I’d bring up the notice thing at this stage as it’s likely to just cause conflict in that discussion. Obviously depending how that chat goes if she is still dead set on leaving ASAP then within the next few days I’d be having another chat with her saying that you understand she wants to leave, it’s unfortunate as you were very happy with her and the children were settled with her, but fair enough if she’s leaving, and you’ll keep her on til she finds what she’s looking for, only please remember you do need and expect and she agreed to 4 weeks notice which is only fair for you as you’ll be left in the lurch childcare wise otherwise. If she says she isn’t going to be able to stick to that then I think I would say in that case you’re giving her her notice now so that you can plan accordingly and not be left feeling like a childcare crisis could descend on you any second. And if she says she will stick to the contract she’s already signed about that and ultimately doesn’t then she’s just a shit person and there is nothing you could do or say to stop her acting completely selfishly. And yeah not a lot you can do although I don’t know how you pay her and maybe I’m a heartless bitch but I’d definitely be thinking if I pay at the end of each month of withholding the last month (or at least some of it) if she leaves without notice for the inconvenience unless she has a very good reason...
I’ve never been financially well off enough to have a proper nanny but I did have a few au pairs when my child was younger and one worked out amazingly for many years, one not so much but nobody’s fault and they acted fairly sticking to notice etc as actively cared about not leaving me in the lurch but just couldn’t cope with homesickness etc. and one was just a selfish person who in hindsight clearly just cynically used me and my child as a “stepping stone” to come to the UK at a time that suited her and a location close to but not quite where she wanted to be, so that she could look for a better opportunity immediately when here in the location that she wanted to be in and with a family who could afford to offer well above the usual pay and conditions most au pairs have and what I could afford to offer her. She quit less than a month in and with no notice, saying she was homesick but I found out that she didn’t go back home but on to this new opportunity locally she had scouted our behind my back. It was a really shit feeling and upset me a lot and she completely left me in the lurch with a serious childcare crisis on my hands as a single parent to the point I had to take time off work at short notice as a GP to try and solve it.