What sort of private care are you seeking? It's not the case that you can just turn up at a hospital whilst in labour and pay for a private consultant/midwife - these things need to be arranged in advance.
I'm local to you. I gave birth to my first at Kingston. Kingston has capabilities for private medical care, St Peter's doesn't. Both offer private rooms to NHS patients at a cost. I was happy with my treatment at Kingston, I was 4cm dilated when I arrived and was admitted to MLU. I have a second degree tear but it was because I pushed when I wasn't having a contraction. Conversely I met a woman on my hynobirthing course who wasn't happy with Kingston for her first labour so was going to St Peters for her second.
Kingston can be fussy about who they accept as a patient. I believe they are one of the busiest maternity units in London. I was living in Raynes Park when I found out I was pregnant and Kingston midwife told me at my booking appointment that I should go to St George's or Chelsea and Westminster, this was despite Kingston actually offering midwife clinics/scans at Raynes Park (where I was at when I was told this!!). If you do want to give birth at Kingston I would self refer ASAP. As pp have said, bare in mind that the traffic to Kingston can be awful from the Weybridge area. I can't say for certain but I would expect that as you are 'out of area' all your midwife appointments would take place at Kingston which could be a drag with the traffic especially if you have appointments in the early morning (all my scans were around 9.30am).
I'm having my second at St Peters. To date I'm happy with their service. I have a lot of friends who have given birth there and they are all happy, I certainly get the impression that there have been significant improvements there over the past few years. My midwife appointments are all at West Byfleet, where I live, which is very convenient. I have the same midwife for all appointments but obviously no guarantee she will be with me when I give birth. At Kingston I had a different midwife for every appointment as they work on a rota system.
I have one friend who has given birth at St Peters and Royal Surrey. She didn't find there was too much difference, slightly preferred Royal Surrey but I think it was because she had an elective c-sec there instead of an emergency like she had at St Peters so it was less stressful.
Kingston, St Peters and Royal Surrey all have neo natal units. Kingston is a level 2, St Peters is a level 3. I couldn't establish from RS website what level theirs is.
In general, the benefit of paying privately is the continuity of care (you'll have the same midwife and/or consultant, give or take i.e. Holidays/illnesses, from start to finish), someone at the end of the line if you have a query, you're often paying for better conditions such as food, ambiance, rooms and possibly the option for a c-sec on a specific date. Unfortunately private care doesn't mean that your baby is safer - The Portland Hospital has a very famous private maternity wing and they unfortunately have had avoidable deaths of babies.
Perhaps a doula may be worth looking into? There's quite a few in Surrey.