It would need to be limited (very) stop to attract pax onto the long distance service. No good if its quicker via London... The other stops would need to join their local service & change where necessary
Chippenham to Bedford via London - 2 hours and 40 minutes (160 minutes) - from journey planner for tomorrow
Chippenham to Bedford via Didcot and Oxford - 1 hour and 40 minutes (100 minutes) - Journey planner to Oxford and then East / West rail projections.
Even if you add in Royal Wooton Bassett and Wantage stops, I can't see it being slower via London. And you could save 10 minutes by taking out the reversal at Didcot.
Note that I've taken a worst case example - journeys to Milton Keynes will make a higher proportional gain. ChrisB - can you give us an example of a journey that would be slowed?