"I booked an award travel on Air Berlin using American Airlines miles and they just changed my flights to new times that are terrible for me. Any suggestions on how to take care of that?"
Schedule changes are common. Airlines constantly update their schedules to adjust to demand, and unfortunately they're generally pretty bad at letting customers know about the changes. However, these can work to your benefit if you are proactive.
In general, if a flight changes significantly (that threshold varies by airline, but generally an hour or so or if you will misconnect), then you should be able either to switch to a flight that works for your needs, even if the flight you are switching to would have required more money to book, or to get a refund for your ticket at no penalty. In the case of using frequent flyer miles, you should be able to get a full redeposit of your miles with no penalty.
Also, your flight may be changed many different times, but if cumulatively the change is about an hour or so different, you should be able to get the free changes or refunds. Always reference the original booking confirmation when calculating the total flight changes.
In your specific situation, you will have to call American. They may be able to route you on British Airways, but just make sure they don't try to charge you extra fuel surcharges. Remind them that the schedule change was not your fault and you shouldn't be penalized. On the other hand, if you book a British Airways award, say with $500 in fees, and you end up changing to an American flight, make sure you get a refund of those taxes and fees. I had a friend recently who was rerouted off of British Airways and it took several times of contacting American, but he got his fees back.
General tips when calling to book a better flight due to a schedule change:
1) Search for alternative options yourself. The phone agent will probably try to reroute you on that carrier so they don't have to pay another carrier, but you shouldn't accept that if the proposed option is less than ideal.
2) Be reasonable. Don't expect to be rerouted on a non-alliance partner - work with the agent to find something suitable.
3) Double check your connection time. If the new schedule violates Minimum Connection Time for that airport, you have more leverage to get a better option. Call your airline to see what their Minimum Connection Time is for the airport you are flying through.
4) If your flight is eligible for a refund, check to see if purchasing a new flight would be cheaper. It may make more sense to refund the flight and then rebook at a lower cost, than having the airline rebook you directly.
