When you login to UCCX/UCCE Administration webpage or Cisco Finesse, the Tomcat cert provides a trusted connection between the server and your browser. For CA-signed Tomcat certs, the trusted connection would be secured once you upload the CA-signed Tomcat cert to the tomcat-trust along with the Intermediate CA cert and Root CA cert. In your case, this doesn't apply to self-signed certs.
Cisco Finesse is a web-based application. Right now, your browser may or may not accept self-signed certs. Do you see certificate errors when you login to Cisco Finesse or UCCX/UCCE Administration webpage? The URL might be highlight red or display a certificate error or unlocked icon.
This wouldn't (shouldn't) prevent agents from logging into Cisco Finesse, however, this might prevent the application from interacting with other applications or services like ServiceNow if they require secure (SSL) connections. Some applications or service might accept self-signed certs as long as they haven't expired. All of this really depends on your company's security policy.
v/r,
Mark
