This is long - sorry!
We looked to move to independent in Y3 for DD1 who had been at a state infant school. She was offered a place, but she did not go because she desperately wanted to stay with "friends" who largely disappeared (she was excluded from friendships, no party invitations etc) a year later and she ended up with virtually no friends at the junior school. I really wished I had moved her. However, she stayed until Y6, did well and passed entry tests to two independent schools, so she went to one of them at Y7. She was the only child from a state school who started there and she was only talking to me about this a few days ago! She has a very strong personality and is very resiliant, but other girls were "bitchy" about where she came from. I had not realised it had been so difficult until 13 years later!
Other DD went to the same infant school and transferred to the junior school as DD1 had. However, the SLT was increasingly dire, no decison about a new Head for 2 years, only just scraped Satisfactory in the old Ofsted ratings so we moved her to the independent school DD1 decided she did not want to go to. DD2 left a couple of friends behind, but people really formed friendships according to perceived social status and this tended to dictate who their childrens' friends were, and were not, until Y6. Obviously we did not fit into any sterotype.
My children could not walk to anyone's house, except one other child in our village who was not allowed to play with my DD1. Never understood why not; although I heard it meantioned that my DD was allowed to wear a denim mini skirt and go to the school disco - so clearly undesirable. We decided it was not worth waiting for the time when the children decided who their friends were so for DD2 we jumped ship early.
When my DDs were little, I had no intention of looking outside the state system. I worked for the LA in Education! However, poor leadership, lack of friends, no party invitations led to a decision having to be taken on the basis it could not be worse and the advantages could be many. As we are in a grammar school county, not far from you, we also thought DD2 would not get to a grammar school. So, we had already thought she may go private and DD1 go to a grammar school. Lots of parents do this around here.
DD1 really wanted to go to her independent school age 11 and was ready to move on from the cliques at her local school and although she got a grammar school place, she wanted to board. So she did. DD2 was quite happy to transfer to a prep school for Y4 and is still in contact with a number of girls from that school and she loved the art there.
From what you say about science and maths grades, if you believe she will not get the grades to get into the 6th form, and the school will possibly take her now; go now! Otherwise you may regret that too. Try and "big up" the extra opportunities there and be enthusiastic. Talk to her about the advantages.