Are you using Lightning Data Service?
If you use Lightning Data Service (force:recordData) The utility bar component will pick up change.

From Salesforce Docs,

Records loaded in Lightning Data Service are cached and shared across components. Components accessing the same record see significant performance improvements, because a record is loaded only once, no matter how many components are using it. Shared records also improve user interface consistency. When one component updates a record, the other components using it are notified, and in most cases, refresh automatically.

Set "recordUpdated" attribute with an action, Then in that action logic, handle refresh view logic.

Example @ here

Are you using Lightning Data Service?
If you use Lightning Data Service (force:recordData) The utility bar component will pick up change.

From Salesforce Docs,

Records loaded in Lightning Data Service are cached and shared across components. Components accessing the same record see significant performance improvements, because a record is loaded only once, no matter how many components are using it. Shared records also improve user interface consistency. When one component updates a record, the other components using it are notified, and in most cases, refresh automatically.

Set "recordUpdated" attribute with an action, Then in that action logic, handle refresh view logic.

Example @ here

EDIT @AjaySFDC @tsalb I think, we can use

<lightning:utilityBarAPI aura:id="utilitybar"/> 

Place it in your Component housed in UtilityBar.

Then, Use onUtilityClick() Method of utilityBarAPI which should register an event handler for Utility click. You can call doInit in that eventHandler logic.

Link 1 Link 2

The above links should help you implement this.

Let me know if it fixes the issue or Did i get it wrong again?