Your problem is that they are all close to the Metropolitan mass that is London, but to get to them you need to go to different stations.
Basingstoke is about 45-60 mins outside of Waterloo station in SW1, central london, while Maidstone is 1hr5 from Victoria, and Warsop are at Chelmsford which is about 35 mins from Liverpool Street.
If you are staying in Walworth, dependent on where you are, your options are Denmark Hill, Elephant and Castle or Oval tube/rail stations. If you are near Elephant and Castle or Oval, then you can go to either of Waterloo or Liverpool Street with little hassle, but it will take you 15 mins in either direction. If you are nearer Denmark Hill, you best get a bus to one of the other two.
Nearest London isn't all its cracked up to be simply because you also have to get across London and deal with whichever Tube lines are closed that weekend.