If all the settings are right and the user has been added to all the Projects, the user should be able to see the projects in the timesheet selection. Have you tried using another browser, is the issue noticed on the alternate browser? If you are administrator try logging in as the users' substitute user and check if the issue is noticed. If the issue is not noticed on an alternate browser or when accessing timesheet as a substitute user, then it probably is a browser issue. Try clearing cache and cookies on the affected browser.