Hi modalo,
To my knowledge, there is no designated tool that could do that with one click.
For Polygon based models:
• You need to enable the axis modification tool [L]
• Then enable snapping to: Axis.
• The (selected) child object in question will stay where it is,
.. but a click into the scene will catch the axis, and it can be moved to the Parent axis.
• Switch off the Axis Modification tool and the snapping.
Please note that all model points are related to their object axis. Changing the axis will create new entries for every single point. If that point information was used somewhere, the value change might impact the setup (e.g., XPresso> Point, or a Point Level Animation)
If your scene has some of these features enabled, test the axis change on a copy first, save early, save often.
All the best
Done