The Turkish law does not prohibit small carry-on bags to be put in the overhead, it only prohibits larger carry-on items with wheels and over the dimension 40x30x15cm to enter the cabin.
However, the ground staff generally does not measure the dimensions for carry-ons and simply does not let bags that could not fit under the seat. I could not speak for BA but I have seen some gate agents at Turkish airports who don't care about the regulation if the passenger has gone through security and passport control without being noticed, and I have also seen occasions where the regulation was strictly enforced and all larger items that made to the gate were forced to be gate-checked.

The normal procedure is that passengers have to check their larger carry-ons at the check-in desk and the carry-on allowance ( for BA -> 8 kg ) should be added to the checked baggage allowance ( including HBO fares ). So, be sure to pack your valuable items into your small bag that you would take into the cabin.