I go on trips which are often 30 minute or more (have been known to go a good hour to hour and a half drivetime away). If a child says they want to sit in a Helicopter, see a Castle, go on a boat, there are not that many places it can be done.
However when my boss interviewed me I made it clear that I was an out-and-about nanny, I like going places and feel it is beneficial for children when they are young... as you can't do it once they are school age as easily (or as low costly... seeing that Under'5s can often get free entry to places).
Road trips can be educational in themselves... when aged 4 one of the children I care for liked Road Signs so much, I got him a book full of road signs which he would flick through as we went on our trips - he now knows a lot of road signs. 2 year old currently loves saying Bridge every time we go under one... and points out things he sees through the window such as animals, diggers and tractors, will sometimes play the game of saying the colour of each vehicle that passes.
As long as your children are getting something out of a trip, then I don't see the trip in itself as being something bad. Children love going on trains, a bus, in fact any kind of transport.
Playdates at the other nannies place of work however I would not feel is appropriate. Far better to arrange to meet at at a place... museum for example... so the children get to socialise and get learn about history, culture.
Meeting up in a park is also quite good... though not sure I'd go on two trains to do that! I do meet other childcarers in a park about a 10 mile distance away... the children can safely walk once in the grounds (Windsor Great Park is rather big, so we do a 1 mile route from a carpark to a cafe and playground) and we have done it since the youngest was a baby. It's a social event plus once they are walking they get exercise as well. It's also exercise for me and a brief time with adults to chat to... as nannying can be a very lonely job.
You need to trust your nanny. It's early days at the moment so that trust needs to be built up. So restricting how far your nanny travels with your children is fine... once you trust your nanny more then consider relaxing the restriction so that your children get to go places where they wouldn't usually go with you. That's one of the good things about having someone else care for them, they learn about things that they may not learn with you. You may not want to take your children to a Castle, on a Steam Train, to a Farm... but your nanny might and your children might enjoy the trip and learn things they don't learn at home. Seeing a cow being milked is quite different to seeing a cow in a picture book.
Consider this... if your nanny had requested to meet another nanny at a Children's Farm... would you permit that trip? As it's early days the answer to that may be No... but in 3 or 6 months time... would it still be No?