I've read your other thread and you sound clear that you don't want this nanny back. It's ok to say that rather than dance around the issue. I get that a lot, I advise small businesses on their employment issues.
I have a nanny too and I don't blame you. I wouldn't be happy keeping her on myself to look after my 2yo.
As she's been with you less than a year she can't claim unfair dismissal unless there's a disability issue. As has been discussed on the other thread, there is a possibility this may be considered a disability, however it's very early days for that.
If I'm honest, in your shoes I would take the small risk of a disability claim and would dismiss her. It sounds unlikely she's going to be fit to work in the near future anyway and as a very small employer you would not be expected to keep someone on the books for months as a larger employer might be.
You could go through all sorts of palaver about writing to her doctor, getting her to see an occupational health specialist to establish the likelihood of her returning to work etc, and a business would be expected to do those things. But I don't think a private employer in your shoes would be - it's about being reasonable and what counts as reasonable behaviour does vary according to the size and resources available to the employer.
There is a risk, but I think I would take it. Write nicely saying that in the circumstances with her signed off sick for a long period of time and in a state where she is not able to communicate with you at all, you don't feel it is likely she will be in a position to return to her job in the near future. You need consistency of care for your child rather than a succession of temp nannies while you wait to see whether she will be able to return and on that basis reluctantly you feel you must terminate her employment. Pay her her month's notice in full, pay her any holiday she is owing, offer to provide a reference to future employers as and when she is ready to seek alternative employment and offer if there's anything at all you can do to help she should ask.
That's what I'd do.