If you are referring to your Bungie.net profile display name, that won't automatically update in case of platform account name change. You'll need to do it manually in your profile Settings > About Me.
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That has already been changed. I changed my PSN account username, but it still shows the old username in the cross save area, as well as on third party sites. Is it not possible to have that changed without deactivating the cross save and re-doing it with the updated PSN account name?