The only way from Hamburg to Battersea is by flight. You should expect to be able to find tickets starting around $335, if you search flights from XFW to LHR. The best time to book shorter flights is 2-3 weeks before your trip.
On your trip from Hamburg 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.