Notify user the doc index has completed

When the doc index build was blocking it was obvious when it was complete.

Now that it's in the background (much improved) you're not really sure when the full docs are available.

The toolbar might be a good place to show some info.

Start by locating the background doc index build and the toolbar entry points in the repository. Determine how completion can be surfaced to the user, then verify that the toolbar provides a clear notification when the full documentation index is available.

Written by the indexing model from the issue text.