Have you looked at the FNRTTC routes? Chelsea bridge, Clapham common, A24 to Tooting, left at Tooting Broadway. A217, then A237 to Coulsdon. Right up Portnalls Lane (a bit hilly), Starrock Lane, High Road, Blackhorse Lane. This brings you to the top of Reigate Hill. Hammer down there, through Reigate, over Cockshot Hill. Pick out Lonesome Lane to get you to Horley. Then B2036 to Crawley. Drop in for a cup of tea at my house. Job done.
Or just follow the A217 from London, filtering carefully as you go.