I don't think anyone should have to risk being filmed getting changed so any nanny would be within her rights to say swimming is off limits full stop.
I think it would be nice if no videoing was actually done in general but all nannies and carers assumed it might be being done and behaved accordingly. I don't know how we get to that situation though. But if I was in that job I'd assume I might be being filmed.
As for the outrage at not being trusted, I'm a bit
about that. All new employers have to go on is references and interview. That's not a lot on which to base unquestioning trust from then on.
The priority really isn't the right of a good employee to feel trusted, if that comes at a price of someone being abused somewhere. OK, if you're a nanny and you know you're a good, reliable person who isn't going to abuse anyone - that's great for you and your charges. But two houses or two streets away or someone not that far away there will be some carers who are not like you and who are going to be neglecting their charges.
Should those vulnerable charges suffer, because it's considered outrageous for employers not to put blind, unquestioning faith in their employee once they've had good references and interviewed them?
Any employee is quite within their rights to turn down a job because they feel self-conscious and uncomfortable about being filmed. I can understand that, and anyone wanting to video a nanny has to be very careful, very open, and is running the risk that fewer people will want to work for them.
Even so, I don't think as a carer anyone can insist "you ought to trust me!" because if I employ someone to care for a vulnerable, nonverbal person, then I actually owe that vulnerable person the higher duty not to blindly trust carers but to check up on them pretty closely and, however I do it, to make sure that those carers are working in an environment where they know that they will not get away with neglect or abuse. So there's a balance to be struck there.