Can Exchange emails be downloaded in Apple Mail to then be uploaded to Gmail account?
If you don't own the account login details, you've picked the best solution, this would be to set up a rule forwarding all new emails to your Gmail email.

Any Mail Client, including Apple Mail, app needs log credentials to establish the connection. So to be able to use Gmail as client app for your Exchange account, you will either have 


to submit those manually  or 
to have the password in iCloud
keychain


For the older emails, you can export them in .pst and upload that to Gmail but that's rather desktop operation. I think this should help.