Speed up search/replace in VBA Word
One thing that is undoubtedly slowing you down is that you are reprocessing found text that is already hyperlinked. You should check that Rng.Text isn't already hyperlinked before you do all that work.

Do While .Execute(findText:=SearchString, Forward:=False) = True
    If Rng.Hyperlinks.Count = 0 Then
        'do your processing here
    End If
Loop


Although that will still perform the search on all of your previous text even if it skips processing them.

If you know that your unlinked text will always and only be at the end of the document, you could use a Boolean flag to stop searching once you hit found text that is already hyperlinked. Something like this:

Dim newText As Boolean
newText = True

With Rng.Find
    .MatchWildcards = True
    Do While newText
        If .Execute(findText:=SearchString, Forward:=False) Then
            newText = (Rng.Hyperlinks.Count = 0)
        Else
            newText = False
        End If

        If newText Then
            Id = Rng.Text
            'add your hyperlinks here
        End If
    Loop
End With


That should speed things up even more.