What model computer? Most monitors like your new one don’t have speakers. No sound on desktops without separate speakers. 

Custom built tower. This display definitely has speakers in it. 

Here is the S2318H User's Guide. The S2318H does have two 3w built-in speakers. It also a rear Audio Line Out port for powered external speakers. Open the S2318H Menu OSD (On Screen Display). Go to the Audio section (page 25). Check to be sure that you have the following selected;

Audio Source = PC Audio
Speakers = Enable

Then go to the PC audio playback and choose S2318H HDMI as the default playback device.

What specific motherboard?

Only if the onboard GPU and onboard audio are setup correctly.

Post a picture of the rear of your tower showing all cabling connected.

Post a picture of the Windows 10 audio playback options.

I have tried this and I have also installed the latest driver for the display but still under sound it says “not plugged in” for HDMI audio 

Sounds like the monitor setup is correct. Post a picture of the rear of your tower showing all cabling connected. What video card model is installed? Or are you using the motherboards onboard GPU? Post a picture of the Windows 10 audio playback options.

Using the on board GPU. Am I correct in saying that I should be able to get audio through HDMI on this model (S2318H)? 

Gigabyte B360M H

It appears that you installed the retail Gigabyte B360M Hb motherboard in our 2011 Dell Vostro 260 case? The Device Manager is showing us that three devices do not have a driver. You need to check with Gigabyte website for chipset drivers. There should not be two audio devices enabled in the Device Manager. Disable the first one, retest. Disable the second one, enable the first one, retest. 

Neither of those worked. Is there an HDMI audio driver I can install?