Yep, 6 of the famous ancient oaks were toppled, it was said Sevenoaks should be renamed Oneoak.
I remember the Kent Messenger newspaper group publishing a book of photos, the scenes were mind boggling, I remember this one of just loads of fallen trees, and the only clue that it was actually a main A-road was a some road signs sticking out amongst the trees.
I grow up in my teens between Oxted in Surrey, and Edenbridge in Kent, 90%+ of trees in that area were taken out. In the video below there's a piece on a couple heading off from Edenbridge on the 45 minute drive to Pembury Hospital, as the wife had gone into labour, they never got there, they were trapped by trees falling both in front and behind their car, it took them about two hours to walk back home, with her in labour!
In this documentary, the on-duty head of the National Grid explains he had to switch off the electric to London and most of the SE, to prevent the whole grid collapsing, taking the whole country out, which would explain why TV and most radio stations in that area were off air in the morning. There would be back-up generators at main transmitter sites, but most would need engineers to reach them and make the switch over, so TV and national radio in the SE probably came back as the morning went on, there wouldn't have been back-up generators at relay transmitter sites, nor stand alone local radio sites.
At a guess, the Caroline newsreader found Radio 4 on long-wave, as that site is in the the Midlands, well away from the power cuts, plus as it covers most of the UK, it's been considered important in times of national emergencies, so was probably manned 24/7.