Instead of building a complete new line. I think if any project of the sort took place it would more likely just be a curve put in it at Yeovil junction leading onto the Dorchester line.
It would result in direct services between Weymouth and Exeter, just not a quick as a new line between Axminster and Dorchester, plus it would be a lot cheaper then a new line
This was raised in the Bristol - Weymouth thread - here's what I posted on that about this suggestion when somebody raised it:
Two words - Business case.
Sorry - but there simply isn't the demand for a direct Bournemouth - Exeter service and certainly not enough to justify the improvements you suggest.
Currently Exeter - Yeovil takes approx 1h 15m. Yeovil - Weymouth 50m and Weymouth - Bournemouth 50m - all in just shy of 3 hours.
The average journey going via Southampton and Sailsbury is 3h 20m.
Even if you improved Yeovil - Weymouth - doubling etc - you'd struggle to save more than 15 mins.
So you're suggesting a massive improvement, for a 30 minute saving to provide a service between two medium sized places. The only places of size served en route would be Poole, Weymouth and Yeovil- that's it.
Weymouth - Yeovil is circa 30 miles - even on a *very* conservative £ 5m a mile to upgrade (bearing in mind signalling etc would need changing) you'd be looking at £ 150m.
Sorry - but it just doesn't stack up.