We have a roughly constructed, movable panel allowing our neighbour and us access to one another's gardens. However, this is by mutual agreement and, even though we're close friends, it does not give us carte blanche to wander in and out of the other's garden at will. I'm sure we wouldn't remain friends for long if that were the case.
I trust you've already spoken to the neighbours about your concerns - the risk to the child as well as your right to privacy. If not, then you should. You need to be quite firm with them, I'm afraid. Tell them that whilst they may think it's OK, you don't, and you'd rather not have their child running around in your garden for all the reasons you've outlined above.
They and their children have no right to be on your property without your permission. Whilst you'd be unlikely to sue the child for trespass or damage, you could argue that she was (or should have been) under the supervision of a responsible adult, and thus it is that responsible adult who must be sued. If no responsible adult, well, you have a whole new approach there. You could also play the 'I'd-hate-us-to-fall-out-over-this' card.
However, you also need to be aware that anyone who has an accident on your property, be they there with or without your permission or knowledge, can legally sue you for damages. It's one of the unfairest loopholes in British law.
At the risk of bringing yourself down to their level, you could put up your own fence or plant a few rose bushes, which they would have no right to touch. But you really shouldn't have to. Could be worth contacting CAB if you have one in your area, or see one of those solicitors who offer an initial consultation for free.