Woking to Hampton will be around 10-15 miles but it is straight up a traffic free m3 and so a reasonably fast journey, I guess the same would be true of Halliford except it is via the M25 which is more traffic prone. Certainly the Woking bus to Hampton is not one of the ones that is notorious for taking an hour plus, those tend to come from London suburbs that get choked with traffic. I think the bus stops at other places around Woking as well including Pyrford and Byfleet.
Swimming tends to be an activity serious swimmers pursue outside the indies, LEH is quite unusual in having a pool, which the boys swim team do use, and that isn't exactly Olympic, 30m? There are national schools galas but with the exception of Millfield I can't think of an indie that you would call a serious swimming school. I think Hampton had an Olympic hopeful though, and LEH a couple. You may also find your expat swimmer may find it a less attractive proposition when it is dark and cold, and they are in some dank slightly mouldy public pool, or at least my delicate little flowers did 
Catholic state schools tend to be highly selective, the good ones at any rate, not on academics but on church attendance, early baptism etc, even whether the parents clean the church
the independent ones though, like St George's make a point of being inclusive, even more 
St James has quite an idiosyncratic ethos, pupils learn Sanscrit for instance, but I know parents who have sons there who say it really succeeds in creating a supportive community and is actually a strength.