Who built the church? An Orthodox Emperor. Who stole it? Roman Catholics and later Muslims. Who does it go back to? DA! This is an easy one, the Orthodox. It is a blessed and consecrated church. It would probably just need a re-dedication, cleansing or something to that effect WHEN we get the church back.
And the money to restore the 1500 year old structure would most likely pour in fast. I'd find extra money to donate.
It'd take a war to get it back, and I dunno if you noticed, Greece has no chance in a war against Turkey, and Istanbul is a megacity, it's not like it would be taken as easily as it might have been 100+ years ago.
If it were fully returned, it'd need a reconsecration, but in this scenario, it wouldn't need to be reconsecrated.