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Hey, @webcart
First of all, the Login Logout Menu plugin isn’t provide such functionality. You can use this plugin to add your Login/Logout links in your nav menu or use the shortcodes for them. Helping documentation.
Let me help you with how you can overcome your issue. It depends on how you apply the protection to your downloadable files.
1 – If you use any plugin, you have to contact their support forum. They can help you with that.
2 – If you did custom work and used the login_redirect hook then the 2nd parameter will help you in redirect back to the same page. Your developer can help you in downloading the file after redirecting back.
Thank you.