The only way from Hamburg to Birmingham is by flight. You should expect to be able to find tickets starting around $346, if you search flights from HAM to BHX. The best time to book long flights is 5-6 weeks before your trip.
On your trip from Hamburg to Birmingham, 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.