Wewishyou - sorry I don't wholly agree with what you're saying. Do you work in employment law or HR?
As a nanny you are an employee, and as an employee with 2 years service or more with the same employer, you have more rights (ie to claim unfair dismissal or constructive dismissal at an Employment Tribunal) than if you have less than 2 years. There isn't any "more or less than 2 years", it's 2 years which is seen in most businesses by management as 'more risky' once an employee reaches that point.
However, to say there is no risk at all under 2 years is incorrect. It isn't just pregnancy or religion to be aware of (in terms of discrimination), but also making a protected disclosure (ie health and safety breaches) and other characteristics like disability, age and race. Whilst the OP has raised issues relating to conduct/behaviour, I always like to know about particular characteristics to weigh up the overall risk of dismissing someone.
In theory, you don't have to give a reason for ending employment with less than 2 years service but a) it's good practice and b) it's fairer and clearer to the individual. In my experience, terminating employment should be fair and clear to the employee - even just to save time and energy later with the person potentially causing you issues afterwards (ultimately to ward off an appeal to the dismissal or grievance).
As a nanny, you're still an employee and should be treated fairly. As an employer, you should be following good employment practices.
If I was the OP I'd be investigating the misconduct first because hearing one side of the story (older sibling version) is not the full picture and there could be an explanation or misunderstanding with what was heard - it could be worse and therefore does warrant dismissal but a thorough investigation is key.
As for not paying notice pay - be careful. There is a big difference between conduct dismissal and gross misconduct dismissal. Nothing about proving the offence happened or not. Only offences of gross misconduct warrant no notice pay so the offence is that serious it warranted a summary dismissal.
Like I said earlier, as an employer, seek professional opinion.