we recently done this in a 9.7 environment and we did it the safe way:
uninstall Workspace UEM on Console and Device Service Server.
Reinstall both Servers and enter the new console URL while installing.
Everything worked afterwards without problems.
There are redirects to console URL in the default.asmx on the IIS and stuff like this. I was not sure if this is changed if you just change it in All Settings > Site URLs, thats why we did it this way.
Other things to consider:
proxy execptions / activation
DNS entry should be there or changes in host files if servers can't get the name via DNS
monitoring of console has to be changed
maybe reinstall of awcm securechannel certificate on device service server (was not needed after our change but could be)
Email Notification Service v2 continued working afterwards. Support told us that we have to send them the cns secure channel certificate after changing the console url for activation on cns server but it worked without that step.