Is it unreasonable for me to ask that my kids only meet up with other children that they/we know or that I have been asked about?
Yes, I think it is unreasonable as it makes things very restrictive. Your children can't make new friends even with children they meet at school, as if your nanny is taking and collecting from school you may never meet their school friends.
How would asking first help? Nanny could ask if it's ok to meet up with other nannies and the children in their care... what would you respond. No? Yes?
Also she impressed upon us the need for planning at interview and there are no activities planned in the diary (or at least written for next week)
Planning things in advance is often the route to disappointment I find. Things can change quickly with children... or indeed the weather. So whilst there may be a general plan of action, it needs to flexible and I feel it's best to accommodate children's wishes as much as possible. Nothing like dragging a child out somewhere they don't want to go. Far easier if you have some buy-in from the child - they want to see a castle... off you all go to a castle.
I now have to call her sunday evening (she's away this weekend) to ask what she's intending on doing with my boys next week as it's the last week of their school hols.
You should not be contacting your nanny over the weekend about such matters. Are your boys happy in her care? That's what you should focus on in my view. Your boys will no doubt moan if they don't like the activities they do.
Does your nanny take photos whilst out on outings? Maybe that could help to show you that your boys are enjoying themselves. Maybe your nanny could text you during the day as to what they are doing, where they are. Doesn't need to be something you reply to... just a text like - We are at Legoland. We are on a steam train. We are at a castle.
How old are your children? You have not mentioned about feeds, or naps, so I don't think your children are that young.
What did you do before you had this nanny? I am wondering if you are finding it hard to let go and trust someone else to care for your children. Trust does take time to build up but unless you give them enough freedom, they can't show you that they can be trusted with that freedom.