Windows

Exchange will not route email that is addressed to a local user outside the exchange server, since you clients are downloading thier email from the hosted pop server (and not you exchange server) it dosn't show up.
If you want to stay with your ISPto host your email and everything else should work, you whould have to stop using POP on the clients computers instead use a pop client on the server that downloads your email to the exchange server. For the clients that have to connect remotely so they can download email. Setup pop on your exchange server.
You did not mention how email is sent from the office out onto the internet. If you use your exchage server to send the email there shouldnt be any problems on this. But if you use your ISPto send the email to the internet you will have to change this as well.

Point value changed by question poster.

My guess is that the exchange server has not been 'told' to send the messages to the outside world. If the mailbox smtp address is the same as the external pop3 address the the MTA will put the messages into the mailboxes. If this is the case the way to send them on from there is to change the internal smtp addresses to something like username@here.local, You can do this at one go from site addressing. Then create custom recipients for each user with the external address, again you can do this at once by creating one, then exporting the directory(Tools,Export) and only select custom recipients, If you then open the CSV file as a spreadsheet you can copy the layout of the first one and change the usernames,smtp address,and X400 address in each subsequent ones. Save the file back as a CSV, then open the same file in notepad and add a <CR> to the end of the text, otherwise the last entry in the CSV does not import correctly.
This done you can import the file into the exchange directory (Tools,Import). If you create a new recipient container for these it will make future editing easier, Then you can set the appropriate Custom recipient as the alternate delivery address for each user. Now the MTA knows that these addresses are external and each one will be sent on to the IMC. After setting up the alternate recipient Hide all the custom recipients from the address book to prevent them appearing in the GAL.

Hope this helps
Chris

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