Hi, welcome to UKCS. It's a bit of an obvious answer and a fair trek from London but there are plenty of campsites like the ones you grew up on in the lake disrict in Cumbria.
If you want to be closer to home then try Sumners Pond near Horsham. Lovely lake and you can pitch right next to it. Need to book early though. Facilities are excellent and site beautifully quiet. - Enjoy.
I looked at Sumners Pond on their website and it appears to be exactly what we are looking for. Thank you very much. If anyone else knows of any simmilar camping spots within a two hour drive from south east London, please let me know.