It would seem that the easiest mechanism for this would be to just create a route which took all of the segments into account. If you create each stop point (Waypoint) in the route and then calculate the overall route, the driving time can be extracted from the Directions collection associated with the entire route or the SegmentDirections collections that are properties of the Waypoints.
See
http://www.mp2kmag.com/mappointqa.asp?p=users&id=533 for more information on navigating the object model for routes.
Once calculated this can be stored in Access or returned to an Access form.
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Walt Cygan