Actually Anne was buried in an old arrow chest. For some reason a coffin hadn’t been supplied for her. Whether this was purely an accidently thing or done delibrately I don’t know. Perhaps it was believed right up and until the end that Henry wouldn’t chop her head off or/and that she would except Henry’s terms and go abroad into exile with Elizabeth. But I do know that that the arrow chest was too small for Anne’s head to be placed where it should be and that it was tucked into a space under her arm. I’ve heard that it was quite common for an executed (beheaded) person to have had their head sown back on prior to burial.
I don’t think it’s likely that Anne’s heart would have been taken and buried in St Mary’s. Basically because her burial was extremely hasty, due to the debarkle of her coffin, and that Henry wanted her out of sight and mind quickly.

I’ve been to this church, it’s rather old and dilapidated, and seen the church organ under which it’s supposed to be buried. Didn’t Anne’s waiting women bury her? would it have been so hard to remove the organ. They weren’t surgeons, so I guess it would be. But how well guarded was St. Peter in the 16th century, exactly? you don’t know what sort of grim souviner hunters might have been lurking around. We need to try not to judge their life and moral codes by our own standards, because otherwise we’re always going to fall short