It depends on the school and how much money you’ve got.
If you’re planning to go all the way from pre prep to A level it’s very expensive.
We went state for primary and private for secondary. The primary was large, lots of high attainers and rated good Ofsted.
Comparing my daughter to her peers (who have been at her academically selective independent all the way through) I would say that she missed out on sport, music, languages, history, breadth of curriculum and depth of teaching, preparation for 11+ test. There was also low level disruption from being in classes with children with SN or behavioural issues. She was also often tasked with helping the teacher teach slower learners rather than being stretched with extension work. I see good and bad points with the latter example by the way.
However, she ‘caught up’ then overtook quite quickly (plus awarded academic scholarship aged 12) as did all of the others who joined age 11. She and other state children have a level of resilience which the private children don’t have so much. I noticed the 11+ joiners were given more positions of responsibility.
On reflection I would probably still have gone state for primary. I think she was let down a bit educationally but benefitted a huge amount socially. Plus the school was a 2 min walk which cannot be underestimated.
However, if she the options for state grammar (we live in no grammar / fully comp area) for secondary then I would have put her in private prep as I think it gave the others an educational advantage.
So it depends what your finances and options are like.