I have recently purchased an LG OLED65CX smart TV and I am unable to download the apple TV app. The app doesn't show on the LG app store.
For a list of specific LG Smart TV models that support the Apple TV app, go to Apple TV app devices - Apple. That includes the LG OLED CX (2020). Hmm.
One thing that might be an issue, if the country or region isn’t supported. Are you in the US?
This model should also support AirPlay. Does that work for you?
Verify that LG software is up-to-date (firmware, WebOS). There was an update available recently (10/13).
Watch on LG Smart TVs - Apple Support
And when I try to plug in my apple TV 4K directly to the TV set via the HDMI port I get a 'no signal' message
That is unrelated. Please verify that you have selected the corresponding HDMI port on your TV for input. Try a different HDMI port as well (you have four). Please verify that your Apple TV box has started up correctly (continuous glowing status light).