Only beers you'll likely find in oak is a Samuel Smiths pub. The beer is average, but the beer is cheap and the pubs can be interesting.
The Market Porter at London Bridge tube/Borough Market is a great stop.
The White Horse at Parsons Green
Jerusalem Tavern has St. Peters ales. It is a really great pub. Well worth the trek!
Those are my top two pubs in London. Otherwise, just wander. All pubs will have real ale. Try the same beer back to back from different pubs. You'll be amazed at the difference. This is beer the way its meant to be.
Look for smaller micros, and don't be surprised at how pale and hoppy English beer has become!