The Bristol/Bath to South Coast Study was commissioned by the Government Office for the South West. Traffic management and directing HGVs via the A34 were amongst the SWRA approved recommendations, which did not include a Westbury Bypass.  The A350 was not considered a strategic north-south route.
These fundamental conclusions have been further endorsed by the South West RSS EiP Panel Report.

Wiltshire County Council, Dorset CC and allies were pestering all concerned to put in upgrading the A350.

The BBSCS was produced by private consultants WPS and its recommendations always had to be approved by the South West Regional Assembly.   Largely on the basis of information etc put forward by Wiltshire CC, the BBSCS included a Westbury bypass as a local improvement (based on WCC's estimate of £14.4M in 2004 - NB compare it with £33M by 2007), although, unlike the route traffic management strategy or area strategy at Bath, neither a Westbury bypass nor A350 upgrades were in the summary of final recommendations.

Here is the February 2004 BBSCS issue for approval [on pdf].  The last page of this final issue of the BBSCS ends with 'Subject to acceptance of these recommendations by the Regional Assembly...'

Here are March 2004 GOSW comments [on pdf] on the reality of the fundamental unsuitability of the A350.

Here is the April 2004 Regional Assembly approval of only specific BBSCS recommendations [on pdf], which, as under item 4, as agreed by all (inc WCC), excludes upgrading the A350 and excludes a Westbury Bypass.

These SWRA minutes also show that there was an attempt to slip in A350 upgrades and a Westbury bypass, which did not succeed, as neither are in the SWRA-agreed/approved BBSCS-based final recommendations.

However, here is a December 2005 Wiltshire County Council report [on pdf], which, as item 24, talks about 'the clear recommendations of the Study which recognise the need for a Westbury Bypass...'     How odd.

And the April 2006 Wiltshire Structure Plan to 2016 [on pdf], by WCC, says, in 5.37, that 'The Bristol/Bath to South Coast Study (February 2004) recommended that the Westbury Bypass... should be proceeded...'

Wiltshire County Council was selectively quoting from the unapproved February 2004 issue of the BBSCS. WCC ignored the April 2004 Regional Assembly which excluded a Westbury Bypass from recommendations.

The SWRA voted for management, for directing long-distance HGVs to the A34 as the north-south route.

Wiltshire County Council prevaricated over it, since 2004, and effectively ignored these recommendations.