Of course they have a right to use toilets.
There are toilets for male and female people and everyone is one or the other. Why can your nephew not use the toilets that have been specifically designated for people of his own sex?
You're not making some big statement about your gender identity when you use one loo rather than another. You're just using a loo. There's absolutely no logic for toilets to be segregated according to gender identity, which is something most people don't even have in any meaningful sense. There is plenty of logic for them being segregated according to sex.
People who identify as non binary have to use toilets which they believe "don't match" their gender identity. I daresay most of them are using the toilets which match their sex. So why can't trans people do likewise?
There are two interesting things here in your post.
Firstly, you still refer to your nephew as your nephew, despite the fact that he apparently identifies as a woman. If you know and love him and believe he is no danger to anyone and yet you still don't believe he is a woman, how can you expect the rest of us to?
Secondly, you say it's not a mental illness. Why do you think this? It's not a physical illness. Prior to receiving any kind of "treatment", trans people's bodies are perfectly ordinary and as healthy as anyone else's. Their dysphoria (if they have dysphoria) is a feeling in their head. A feeling which apparently makes them so miserable that they want to change everything about who they are, even to the point of having irreversible medical interventions which harm their physical health but may allow them to superficially resemble a member of the opposite sex. How is that not a mental illness?
I think it very much is a mental illness, and I cannot think of any other example of where we harm the physical body to try to make it "match" what is going on in the person's brain, rather than attempting to actually treat the mental illness.
Personally I think it cannot be beyond the wit of the pharmaceutical industry to invent a drug similar to an antidepressant which alleviates the symptoms of gender dysphoria and allows someone to live more comfortably as the sex they actually are. But there wouldn't be as much money in that, would there? Why would they do that when they can sell that person a lifetime's supply of cross sex hormones, plus all the other drugs they are likely to need as a result of any surgeries they have?
This is basically a global racket by the for-profit medical industry. A lot of people are getting very rich off the back of vulnerable people like your nephew. They do not care how much harm they do, and they will not stop until the medical malpractice lawsuits start to pile up and insurers stop providing coverage for this sort of "care".
Anyway, I have digressed.
However sad and vulnerable your nephew is, his needs don't trump those of women and girls.
There is no reason why he can't use men's toilets.