If I understand you correctly, the setup is follows:
One server is visible to the internet (= exposed server).
If a client from the internet connects to this server, the server in turn connects to other servers (also running apache) in your local network.
And your problem is, that a www connection from the (exposed) server to the (other) servers does not work.
Even if you can ping the other server, a firewall may block (www) connections from one computer to another.
Perform the following test:
On the exposed host use telnet
to connect to that server on port 80 (www).
telnet ip_of_other_server 80
If the connection is refused immediately, a firewall is most likely blocking that connection. The other possibility is that the web server on the other
computer is not running, or not running on port 80.
In case a connection is established enter
Then press Enter two
This requests the root page from this web server. If you get some kind of response, then that web server is running.
You may repeat the telnet command on the other
computer. In this case the web server and telnet command run on the same computer (with no firewall blocking them). If this does not work as well, you should examine the web server on that computer.