Are you being shown the question: “Cannot connect any display running. Do you want NoMachine to create a new display and proceed to connect to the desktop?” right? Can you choose “No” and wait to see if the session appears on the session list?
NX server logs from server side would be helpful to investigate problem. Please enable debug, reproduce problem and send logs to forum[at]nomachine[dot]com. There is instruction how to enable and collect debug logs automatically: https://www.nomachine.com/DT10O00162
In the meantime is there a way to restart the NX server on a machine either remotely or via the command line that I could use next time this happens?
Command for restart nxserver is
sudo /usr/NX/bin/nxserver --restart, you can also execute it remotely using SSH.