I can clarify the dismissal front for you: you can make someone redundant within 2 years without having to pay them redundancy terms (though you must honour their notice period)
You should only dismiss someone for a 'fair' reason. However, if she has been working for you for less than a year then she cannot take any action against you even if you have not given a fair reason. You would only be at risk of any legal comeback if you dismissed her for a discrimnatory reason.
Anyway, given that she has 'frustrated' the contract by not complying with the agreement to become ofsted registered within a reasonable time then dismissing for this reason would probably still be ok even after a year.
If you do decide to dismiss you must still give a notice period (or pay in lieu) although I would think it perfectly ok to withold the cost of ofsted fees from her final check if she hasn't used the monies for the intended purpose.
In terms of what I reckon you should do, no particularly strong feelings one way or another: yes she is messing you around, but some people are just flakey like this - doesn't mean that they aren't great nannies. FWIW my current fab nanny changes her mind constantly about whether or not to move north to live with her boyfriend - it has been going on for months now, and I have learnt not to stress about it - when she leaves she leaves, I will stress then. Apart form that, it is her life, and I am just grateful that we have had her for far longer than I hoped.
BUT I don't understand the registration thing at all. I would be concerned that there may be an underlying reason why she is avoiding this.... why not tell her she doesn't have to pay it back, and see if she then jumps to get it done? If she doesn;t then I would suspect bigger issues (maybe she has been arrested recently or something?)