Then you'll need to store the last Initials value, either in a cell or in a named range. You can then use a Worksheet_Change event to compare the new value of the cell with the last set of initials chosen, and update the links if required. Something like this:
Const cstrWORKBOOK_NAME As String = " Task Planner.xls"
Private Sub Worksheet_Change(ByVal Target As Range)
    Dim arrLinks, i As Long
    Dim strOldLink As String, strNewLink As String
    ' check if A2 changed and if it's not blank
    If Not Intersect(Target, Range("A2")) Is Nothing And Len(Range("A2").Value) > 0 Then
        ' get new initials
        strNewLink = Range("A2").Value
        ' get last initials (need to strip off leading = sign)
        strOldLink = [LastInitials]
        If StrComp(strOldLink, strNewLink, vbTextCompare) = 0 Then
            ' no change, do nothing
        Else
            ActiveWorkbook.Names("LastInitials").RefersTo = strNewLink
                
            ' find the link we want to alter
            arrLinks = ActiveWorkbook.LinkSources(xlExcelLinks)
            If Not IsEmpty(arrLinks) Then
                For i = 1 To UBound(arrLinks)
                    If InStr(1, arrLinks(i), strOldLink & cstrWORKBOOK_NAME, vbTextCompare) > 0 Then
                        On Error Resume Next
                        ' turn events off so we don't get stuck in a loop when the formulas get changed
                        Application.EnableEvents = False
                        ActiveWorkbook.ChangeLink arrLinks(i), Replace(arrLinks(i), strOldLink & cstrWORKBOOK_NAME, strNewLink & cstrWORKBOOK_NAME)
                        ' turn events back on
                        Application.EnableEvents = True

                        Exit For
                    End If
                Next i
            End If
        End If
    End If
End Sub