Display not working with Mac mini
Some Mini DisplayPort to HDMI adapters won't support HDMI at higher resolutions, it is possible this adapter won't work at the higher resolution of your new monitor.

At work myself and my colleagues have found this is the case with a lot of cheaper adapters I used to buy, such as this one:

https://www.startech.com/AV/Displayport-Converters/Mini-DisplayPort-to-HDMI-Cable-Adapter-Converter~MDP2HDMI

You could try a new adapter, these are the ones we buy at work and they seem to work well:

https://www.amazon.co.uk/Griffin-GC17096-Video-Display-Converter/dp/B003CK8TZ8