The MQTT communication between publisher and subscriber happens through an MQTT broker. MQTT publisher sends (publishes) some information to MQTT broker on a specific topic. Subscribes then asks for information on that topic from MQTT broker.
Your MQTT publisher is connected to the LAN network of the router. So it can send any traffic to the internet, as the router allows all LAN packets to WAN. Hence, your publisher device can send topics to the broker.
The commands I have provided would let us know if there is any communication on port 1883. When you start 'tcpdump port 1883', you will see if MQTT packets are coming in/out of the device and the IP addresses associated. After you start the command, please, try to publish some topic from the publisher to generate some packets. Do you see any packets? You can take a screenshot and send it to me via private message.
If you are having problems, please, generate a troubleshoot file. Troubleshoot file can be downloaded from System -> Administration -> Troubleshoot. Attach the troubleshoot to your question or send it to me via private message.