The only way from Köln to Battersea is by flight. You should expect to be able to find tickets starting around $299, if you search flights from CGN to LHR. The best time to book shorter flights is 2-3 weeks before your trip.
On your trip from Köln to Battersea, it is best to book as soon as possible. It won't necessarily help reduce your price by a large amount, but will ensure that you have a spot on the bus or train.