I've known myriad Heathers and never known one that went by a nickname. I think that
Heather is one of those "nickname-proof" names.
Oh, wait. I just remembered. I have a 40-year-old niece named
Heather. When she was very young, 1-3 years old, her father would sometimes call her "Hezzy." Not all of the time. And I remember that my mother hated it. And it was not at all something that stuck, he was the only one who did that and even he stopped doing it once she grew older. But I suppose Hezzy is a possibility.