I have a DD who is a beginning of September 2020 baby. Some would also probably consider her advanced (knows letters and sounds, writes part of her name, beginning to understand addition). I consider her a curious child who learns what interests her. I've never pushed it.
Her friends who are 1 and 2 weeks older will be going to school in September and I have no doubt she'd academically be fine doing so. Emotionally though, i'm not so sure. I'd never dream of pushing for early admission.
I've never understood the british rush to school. It used to be nearer 5 and now seems to be pushed to 4. You need to consider more than just academic learning. Also consider her future. GCSE years, university years. Whats the rush? i realise that my DD will be 17 and 18 before her peers at school. She'll potentially be the one buying the drinks and driving them all around while they wait to age. No issue with that.
I get that some children may tire and be somewhat limited by nursery by the end as they near 5. But we balance learning at nursery with home. It's my responsibility as a parent to also support her learning. Museums, activities, reading etc. We also have a great nursery generally, who have a QTS in their preschool room and they support and encourage her love of learning. If you find nursery is limiting her look at local preschools or even private preschools. Or just accept 3-4 is not an age to worry about academia!
I do get your concern. I am also a September born and got incredibly bored at school by 8. But as a parent if this happens for my DD i'll look at how to encourage learning outside school. Tutors, extra criculars, family learning, flexi or homeschooling are all options. But i'll deal with that if and when it happens. I'd rather that than a child who's academically and emotionally struggling and i caused it by pushing her too early. Embrace the play at 4.