Yes I agree with @LIZS – the key factor may be the school year of your DC.
State schools are unlikely to allow them to be in the wrong year. Private schools may be more flexible. When is your 14yo's birthday (ie when does she turn 15)? If it is before Aug 31 this year, she would already be in year 10, which makes a move very problematic.
If she is older in the year, and turned 14 since Sept 1 2025, then she would be in year 9 now. Ideally you would move before the end of the summer term to get her into the end of year 9 and able to decide on options – tho the reality is most students are picking options this term, so there might not be much choice. But moving any time after the start of year 10 is a bad idea.
Yes also – when is your 11yo 12? Again, if that's next school year (after 1 Sept 2026) then at least if you moved this summer he would be starting secondary with the rest of the cohort, in year 7. But you have missed the application window for that so you would be a later application at best.
Overall, private schools can be more flexible on these issues so I would ask the question of any private schools you are considering, depending ofc on the birthdays of your DC and when you are planning to move.
In terms of “is private school better”, obviously it really depends, on the schools and your DC. There is another thread rn about a parent who sent one DC to private and the other insisted on state – and seems to be doing well and be happy. So it’s not one size fits all.
Obviously I don't know about your finances but be aware that private school, even if only secondary, for three DC is a big £££ commitment. You are looking at at least 10 years of fees (assuming they go state for sixth form) which could cost more than £200k (conservatively estimated) all told, plus extras and trips and uniform. That may not be an issue for you of course.