Trains were running in a number of cases, just few and far between. From how it looked (without the insider knowledge of actual trees down and whatever), SWR just couldn't be bothered. Where there were issues, Open Train Times had it marked on the maps where there were issues and you could see the line was open as a few trains were running).
Hampton court branch was open, services were random services which ended up being Dorking trains diverted. Why were they in place when the advertised timetable (for today was only for Surbiton - Hampton Court shuttles. Just run the shuttle.
Shepperton and Chessington seemed open all day. Services were random and plenty of delays though on those which ran. Generally, no delays on route and turnarounds at Shepperton and Chessington seemed decent, just seemed to be Waterloo causing delays and it seemed not wanting to set swap so that some of the delays could be reduced. Kingston to Barnes, line open, no trains. Hounslow Loop, line seemed open, no trains (trains advertised as running to Weybridge but I think there was a tree blocking the line towards Staines but the service could have ran Barnes to Staines).
When the lines reopened through Weybridge, the service was extremely limited.
Just seemed to be a case of 'We issues a do not travel warning, you want to travel, that's a you problem, we told you not to, we don't need to provide you with a train'.