Once your flight is officially cancelled (and no sooner!!) they are obliged to offer you rebooking on a different carrier. Your flight needs to be officially cancelled though.

Also, if BA is confident that they themselves can fly you there on the 23rd, it has no obligation to rebook you on an other carrier for that day. I would be surprised if they rebooked you for that day now.

Last but not least, flying LHR-IST via AMM is a serious reroute and won't be considered a reasonable alternative.