Picking a time based on when the app was last used
Whether you use ternaries or classic if/else you can often make your nesting easier to read by inverting your conditions. i.e.

! window.localStorage
    ? console.warn('Current browser doesn\'t support Local Storage API');             
    : NotificationsStorage
    &amp;&amp; NotificationsStorage.lastCheck 
    &amp;&amp; NotificationsStorage.lastCheck &gt; today
        ? (
            this.lastCheck = NotificationsStorage.lastCheck,
            localStorage.setItem('NotificationsStorage', JSON.stringify({ lastCheck: now }))
        )
        : (
            localStorage.setItem('NotificationsStorage', JSON.stringify({ lastCheck: today })),
            this.lastCheck = today
        )


Although you can't do it with a ternary operator, I also usually prefer to use early returns to limit nesting. This also avoids the issue that you access localStorage in your first line even though it might not exist.

if (! window.localStorage) {
  console.warn('Current browser doesn\'t support Local Storage API');  
  return;
}

const NotificationsStorage: { lastCheck: number } = JSON.parse(localStorage.getItem('NotificationsStorage'));
const today: number = moment().startOf('day').valueOf();
const now: number = moment().valueOf(); 

if (NotificationsStorage
    &amp;&amp; NotificationsStorage.lastCheck 
    &amp;&amp; NotificationsStorage.lastCheck &gt; today) {
      this.lastCheck = NotificationsStorage.lastCheck,
      localStorage.setItem('NotificationsStorage', JSON.stringify({ lastCheck: now }))
 } else { 
      localStorage.setItem('NotificationsStorage', JSON.stringify({ lastCheck: today })),
      this.lastCheck = today
}


You could simply that last if by separating the variables and the logic to update them:

const {newStoredValue, newLocalValue} = 
    NotificationsStorage
    &amp;&amp; NotificationsStorage.lastCheck 
    &amp;&amp; NotificationsStorage.lastCheck &gt; today 
    ? [now, NotificationsStorage.lastCheck]
    : [today, today];

 localStorage.setItem('NotificationsStorage', JSON.stringify({ lastCheck: newStoredValue }));
 this.lastCheck = newLocalValue;