It's a myth!!! Trout streams with wild brownies and cheap near London??!
I came here 7 years ago with same passion and enthusiasm but after years of researching and travelling I gave up. My first places for that are Scottish borders and Wales.
Southern England is full of Chalk Streams and rivers, unfortunately all of that faces the pressure of millions of fresh water fishermen and fact that almost 95% of waters in UK are privately owned gives you no chances. In my personal opinion as I work in the financial sector, the proximity to London has made the situation unsustainable as most of the rich fella that can afford £250/day are coming from the city.
If you want have fun close to London (you can get there by tube+bus) you can go to the River Wandle, it's an urban chalkstream river and it's free!
There are few Browns but the most frequent encounter is this lovely fella

You can foul them on a dry with royal wulff or palmers and you can encounter some big buddy in the deep pool with bloodwarm or hydropsyche larvae
Otherwise marxach advise above is top notch for Graylings