Wycombe Abbey, Downe House or Westminster?

I guess it depends what you want. I went a long time ago to DH. Had a great time. Nice house system, sports, drama. Definitely don’t need to be a genius, but a good number went to oxbridge. Don’t really know what things are like more recently but I think headmistress is the same.

I fully agree with the above poster.
I know from first hand experience that these schools (WA is the one I am familiar with) are not remotely as selective as most London independent day schools (and I am not just talking about the big names). They have a huge inflow of very hard working girls at 13+ or 6th form that will boost their results and propel them at the top of the league tables which creates a "hype" and the total misperception that you have to be a genius to be accepted there. This is not the case at all - certainly at the 11+ stage.
Don't get me wrong it's a very good school but I would not get blinded by the results and do a bit more due diligence to understand how they get the results and if it's the right environment for your DD.

Bright and eager to learn, yes.

In my day IQ tests were fashionable and the average score was 120 - high, but by no means out of the ordinary. Long been discredited but it will give you an idea.

Lots of girls try for all three WA, DH and CLC and all meet up with each other at the open days, assessments etc. They are very different schools, though. CLC very much in the town. WA described itself to us as "Narnia", in that the town doesn't really know what's behind the walls. DH is absolutely in the middle of nowhere.
Academically WA is the hardest to get into, then CLC and then DH.
WA also seems to have a lot of girls from London, and I was concerned about it emptying out at weekends, which is fine if you are from London or are happy for something more akin to weekly boarding. CLC has a London contingent, but equally country and the impression I got of DH was that it was much more country. Obviously all the schools have girls from all over the world.
They are all excellent schools and I echo a previous poster who said get your DD into a good prep school. Once it is possible again, I would recommend visiting the schools, both with your DD and without as it's easier to ask the annoying questions without DD. And remember, if you are looking for 13+ entry you will be trying to pick a good fit for teenage/young adult DD rather than child DD. All of the schools will want to see academic achievement and a variety of extra-curricular interests, boarding schools like them to be busy.

There is no one school that is right for every girl it is tricky to find the best fit for each individual girl. Additionally the right school at 13 is not necessarily the right school at 16. Strengths change and develop.

The assumption that you have to be a “borderline genius” has really made me giggle.

I attended two of the schools mentioned above. Absolutely nonsense to suggest you have to be “elite” or “the brightest and the best” to get in to either.

If you're looking at WA and Downe at 13+ you could also consider Cheltenham Ladies, St Mary's Calne or Benenden (also 11+ entry for all).

I’m assuming you want 13 plus entry. This adds co Ed boarding schools to the list of possible schools. Eg Rugby, Oundle, Uppingham etc.

DD was at Godstowe and plenty of the girls are smart. However it’s not just the scholarship set that do well. You would need to be bright for WA but you don’t have to be nailed on Oxbridge. Godstowe is famously all ability. Plenty of DDs go to less academic schools after going there but they still do well. You really don’t have to go to the top name schools to be happy and achieve well.

You do not need to be a genius to get into WA or DH. DD's friends went and they certainly were not geniuses.

If you are mentioning these schools, you are presumably not interested in State schools.

Your best plan is for DD to come here much earlier and enrol at a Prep school that has a track record in preparing girls for these schools. Godstowe Prep is a boarding girls' prep to age 13 in High Wycombe and that is the obvious one for Wycombe Abbey and Downe House. Many girls have gone to these schools from Godstowe at 13. I think they would also give you a much better idea of what other schools might suit. Far better than googling anyway.

You are also ludicrously dismissive of Downe House. Its perfectly good academically and no one could possibly be sniffy about them. However, it is not London. It is rural. So are lot of boarding schools! High Wycombe is not any great shakes as a town either.

I think that, if your DD is being taught in another country, you need experience of the curriculum here in order to get into the highest achieving schools. But to achieve highly at 16 or 18, you do not need to be at one of these. Many parents look for other attributes in a school to suit their DC. I suggest you do the same.

My issue is that you are listing some of the most elite schools in the country. Your daughter will need to be a borderline genius to get into these and if you get her to pass the test and she is not that standard, than she will hate it.
There are thousands of students that get into the top universities from normal state schools. As long as you find a balance in making sure she is happy and well supported, without being overbearing, she will be fine and get in where she needs to go.

Every school is different. You cannot simply say school A is better than school B. If you just want to compare academic aspects, you can refer their A level results.

Isn't St Paul's a day school?

OP You would get more help if you gave us more info, e.g. current age of your DD, whether you are looking for full boarding, flexi boarding, weekly boarding or a day pupil.

What are you looking for from a school, what extra curricular do you think is important, do you care if a high % come from one particular overseas demographic, etc.

I don't think the best ones have better atmospheres.....you're clearly just googling the best schools. If you're looking at sixth form surely this will be mostly your daughter's choice?