I recommend to manually open the Excel file that the LINK-fields in word refer to before you open the word document.

At least in my case with 95 fields it made them update instantly.

In case the problem remains you could try writing a procedure, that reads the excel sources from the link, opens excel, collects the values and directly writes them into the respective Fields().Result.Text - this way you can skip updating the fields but still change their displayed value and the LINK as such also remains.