I wouldn't think it's likely you'll get compensation. Typically with a major schedule change you can get a full refund (even if the fare is non refundable), or take the reaccommodation option.

I would check to see what other *A options might be available rather than the AI/BA flights. The automated system or the person who rebooked you may not have given you the most direct option.

The difficulty with asking for compensation in this case is the number of carriers involved, and the fact that you booked through lastminute.com. The airlines will likely push you back to lastminute, and lastminute will likely try to deflect any customer service requests. I think it's unlikely you'll get AC miles for a flight cancellation on BD, even if the BD segment in question was part of an AC flight. And it's unlikely BD will compensate you, since your ticket was booked with AC.

In short, I wouldn't spend too much time trying to get compensation. Look for other flight options that better suit your needs, but also be happy that the airline was able to find something that didn't involve changing dates!