Oh mighty. She does sound like a nightmare.
It's completely and utterly unreasonable, although I bet she's probably thinking the same of you as you didn't stay to help (not that I'm saying that she's right of course, she's completely in the wrong here).
It seems like you're going to go round in circles here - you'll feel understandably fed up because they're not treating you with any respect or giving you the very basics. This means of course you're not going to want to put yourself out and so she'll feel more stressed and be even more of a cow.
I expect at the moment trying to sort you out is making her more stressed, even though it's definitely not your fault - it's something that she HAS to do as the employer of a nanny.
Is there any way you can approach the dad and ask for these things to be sorted out? I think if they knew that you wanted to go back to London at the weekend because you were bored catatonic then it would help explain why you need things and also that you weren't being unreasonable.
To help things along can you try to deal with as much of it yourself - eg suggest that they give you the money to buy a cheap portable and that you'll get it. You could also do with a portable DVD player with a screen for Dorset (maybe don't ask for all of this at once). Perhaps ask permission get your own lock and put it on (although I realise that the lock isn't the deal breaker here ).
Re the hot water you should be able to find an over-ride button on the boiler to get the water on (although be careful to toggle it back off so that the usual water cycle continues or they might get back with the babies to cold water)
You may find that once everything is sorted out and the mum isn't pg it will be a good job, but you do need to have the boundaries set. If you think you can't hack it any more or the mother turns out to be a psycho-bitch (equally possible) then you'll have no problems finding other work in Kingston/Richmond. Richmond is nicer than Kingston btw.
If all else fails, I hear the Beckhams are looking for a nanny