One option would be to simply send an email to the user saying "There is a new timesheet waiting for approval". In the email, include a link to the form library.
Promote a column in the SharePoint form library with the status. When the user visits the form library, the status column will clearly show which items are pending approval. Basically, you want to get the users accustomed to do their work in the form library and not via email. It will take some getting used to, but it's the easiest option.
I don't believe that you can include a link in the email that will resubmit the form to SharePoint whilst changing the status.