Well, it's a bit of a detour, but whenever I go down from London to Southampton, I head for Godalming, then take either the 286 or the 283 south to the 272, which'll take you as far as winchester and is generally less busy and less caravantastic than other routes Nice and sweepy too. Much more fun than the motorway, which I avoid at all costs!
Happy riding. May see you on the route!