openUtility() from Utility API can be used to auto-pop Salesforce Omni-Channel Widget.

1. Create the following Lightning Aura Component.

Component:

<aura:component implements="flexipage:availableForAllPageTypes" access="global" >
    <lightning:omniToolkitAPI aura:id="omniToolkit" />  
    <lightning:utilityBarAPI aura:id="utilitybar"/>
    <aura:handler event="lightning:omniChannelWorkAssigned" action="{! c.onWorkAssigned }"/>
</aura:component>

JavaScript:

({
    onWorkAssigned : function( component, event, helper ) {
        
        let workItemId = event.getParam( 'workItemId' );
        let workId = event.getParam( 'workId' );
        console.log( 'workItemId is', workItemId );
        console.log( 'workId is', workId);
        let utilityAPI = component.find( "utilitybar" );
        utilityAPI.getAllUtilityInfo().then(function( response ) {
            /*
             * Following Code will open the first Utility. 
             * Add Omni-Channel as the first Utility.
            */
            let myUtilityInfo = response[ 0 ];
            utilityAPI.openUtility( {
                utilityId: myUtilityInfo.id
            } );
       })
        .catch(function( error ) {
            console.log( 'Error occurred', JSON.stringify( error ) );
        });
        
    }, 
    
})

2. Add the Lightning Aura Component to the App Utility.
Note:
Make sure Omni-Channel widget is the first Utility. Also “Start automatically” should be enabled.

3. Test it by assigning a work to an agent. It will auto-open the Omni-Channel Widget.