When you use standard hyperlinks it hard codes the current workbook/sheet referencing with it.
You could use hyperlink formula instead so that is dynamically changes with your workbook name. e.g. =HYPERLINK("#Sheet1!"&"$Target Column$"&Target Row,"Link Name")
I just tested a workbook that I have with the above formula, when I saved as a new workbook name the hyperlink still works correctly.
Just the sheet reference & target sections of the formula to suit each cell you are linking to.