I may not be of any help, but here is a suggestion. Based in the info you provided It's hard to diagnose what the issue may be, but my best guess is that you are connecting to a POP server. Once you connect to the server via your computer and recieve the mail, (and subsequent mail if you stay connected) the server knows it has sent your mail and will not resend it to the computer, or any other device, like your phone, till you disconnect and reconnect with with another device. Then only new mail gets sent.
This all depends on if you are connecting via POP or IMAP. POP sends your mail to the device (your computer) and deletes it from the server. IMAP leaves the mail on the server so it can be read from mulitple devices. With POP servers there is usually a setting where you can chose to leave the mail on the server after it is initially recieved, so you can POP it to multiple devices. If you are connecting via POP you will need to find this setting and change it.
Again, this is just a guess based in the information you supplied. Good luck.