I'm an employment lawyer, so I'll give you my brief professional opinion...
The first point is that an employee does not accrue the right not to be unfairly dismissed until they have acquired 1 year's service. In this case, your nanny does not have this, so, on the face of it, she would not have the right to bring the claim and you could simply dismiss without providing a reason or following a fair process. However, there are limited circumstances in which employees can bring claims for unfair dismissal without a year's service (I don't have time to list them out here but if you Google it I'm sure you will find it somewhere). I doubt, however, that any of those exceptions would apply. But, the other thing to be very careful of is claims for discrimination, for which no length of service is required. Potentially, she could argue that she was dismissed for a discriminatory reason (she would obv. have to identify such a reason and have a reasonable case) and bring a claim on that basis. Damages would be assessed in must the same way as for unfair dismissal plus she could get an injury to feelings award.
So, for those reasons, it is usually a good idea to try and identify a fair reason for dismissal and follow some sort of fair process prior to dismissal, even where an employee has less than 1 year's service.
You had performance issues (household duties, cooking, general attitude etc) with her that you haven't yet addressed via a formal process. Accordingly, these matters shouldn't really form any part of your decision to dismiss on this occasion. In order to dismiss for poor performance, you should notify the employee of the problems, discuss the issues with them and give them a reasonable opportunity to improve.
In fact, your grounds for dismissal are nothing to do with her performance. Instead, it is the following:
(1) Her failure to follow her employer's reasonable instructions (to pick your child up from school at a certain time) without any reasonable excuse e.g. if she had been in an accident or something and couldn't get there then obviously it wouldn't be reasonable to sack her for failing to be there.
(2) The consequences of that failure, potentially placed your child in danger. Okay, he had his homework club, but, how was she to know he would go there? He could have wandered off anywhere really. You are paying her to take care of your child between certain hours and she failed to do this without good reason.
(3) Furthermore, she compounded the issue by failing to report it to you. This is arguably a dishonesty offence (although there is some caselaw which says that employees are not obliged to report their own breaches of contract to their employers).
In my opinion, this amounts to a serious breach of the implied duty of mutual trust and confidence that exists between employer and employee. I would be fairly confident that her behaviour constitutes a "repudiatory breach" of contract i.e. so serious it permits you to terminate. However (disclaimer alert!!) I obviously don't have all the facts and am basing this on what you have said, so, to be absolutely sure of your position, you really should consult a solicitor or a CAB before dismissing her. Okay, it may cost £150 or so but it would be money well spent to have the confidence in yor decision.
In terms of process, I would suggest you write to her setting out the the "allegations" (not the performance issues as these are not part of the dismissal decsion). Invite her to a meeting to discuss. Allow her to put her side of the story - you need to bottom out the issue of (i) was there a reasonable excuse for not being there (I know she has said there is none but you need to establish this "formally"); and (ii) why she didn't report the incident immediately too (this will be much harder for to explain away). Then tell her you need to think about it and you will let her know your decision shortly. Then write her a letter confirming her dismissal. You do not need to pay notice for gross misconduct offences. However, she may have a go at you for wrongful dismissal (i.e. try to claim her notice money at a Tribunal aruing that the offence did not constitute gross misconduct and she was therefore entitled to her notice money - she can bring this claim regardless of her length of service). Therefore, you might be minded to just pay her her notice money to get rid of her relatively quietly.
Whilst all of this is going on you could potentially suspend her on full pay (although it would be safer to do this only if your employment contract allows - if it doesn't then there are risks associated with doing so, although, if you have really lost faith in her, then (speaking as a mother rather than a lawyer) I would take that risk.
Hope that helps.