It was a combination of factors that preserved Bath’s Georgian core during the period of postwar redevelopment.
To begin, at the end of the Second World War, the urban core of Bath had one of the highest concentrations of Georgian architecture in Britain. With the passage of the 1944 Town and Country Planning Act, a list of structures throughout Britain to be preserved was compiled. Bath had 3,000 buildings on this list; 95 percent of these were Georgian. So even though 19,000 buildings in Bath were damaged or destroyed by bombing during the so-called Baedeker Blitz of April 1942, there were still plenty left to give the urban core of Bath the strong Georgian feeling it still retains. Added to this, much of the architecture of the Victorian period, though generally not neo-Classical in style, used the same primary building material: the golden, locally quarried Bath stone. This gives the city an overall impression of uniformity despite variations in architectural style.
Bath had a strong modern preservation movement in place even before the war, with the Old-Bath Preservation Society established in 1909 and the Bath Preservation Trust in place by 1934 (the national organization The Georgian Group followed in 1937). Though statutory powers of preservation were limited in this period, the influence of non-governmental organizations was enough to restrict proposed changes to existing buildings, traffic patterns and housing that would have significantly altered the appearance of Bath. These groups grew even stronger in the postwar era when public interest in preservation expanded.
The city council of Bath was also aware of the historical and artistic importance of the existing architecture early on, wishing to balance economic revitalization based upon the expansion of trade and commerce with development of the already well-established tourism industry. Though changes to the urban fabric of Bath were proposed as parts of national legislation, local authorities were able to resist control from Westminster and shape the adoption of redevelopment plans. This allowed Bath to restrict creation of new motorways and housing estates to the periphery.
The planners who prepared the redevelopment studies for postwar Bath were also sensitive to the existing architecture. Patrick Abercrombie, author of the 1945 Plan for Bath, was, like many modernists, appreciative of Georgian architecture for its uniformity, symmetry and structural rationality—characteristics it shared with the functionalism of the twentieth century. This led him to suggest preserving much of the urban fabric with only selective removal of (in his opinion) unattractive Victorian buildings and some less-important eighteenth-century structures he classed as “Utility Georgian Buildings”. His plan called for the creation of grand axial views and new institutional buildings in a neo-Georgian style, though little of this was realized. This is in contrast to the plan he prepared for postwar Plymouth, where the devastated city center was entirely rebuilt in a modernist style.
What buildings that were proposed for the postwar city center of Bath were largely scaled to fit with the existing urban fabric in terms of height, volume and traffic patterns. Many proposals were only approved when modernist exposed concrete was replaced with Bath stone cladding. This is interesting because it comes well before the development of contextualism in postmodernist architecture, when architects selected building materials to complement neighboring structures. Bath has a long history of “façadism”, where uniform exteriors conceal a broad variety of interior arrangements. A famous example of this is the Royal Crescent, known for its "Queen Anne fronts and Mary-Anne backs”.
But Bath didn’t entirely escape the 1950s and 1960s without alteration. Some notable examples of postwar modernism in the city include the Bath Technical College, the campus of the University of Bath, the (now demolished) Southgate Shopping Centre, and the Kingsmead Telephone Exchange.
To respond to your 'but', you might like to consider Exeter. First, if you aren't already aware, look at the 'Baedecker Raids', of which Exeter was the first recipient. These destroyed Bedford Street/Circus - see http://www.exetermemories.co.uk/em/_streets/bedfordstreet.php as well as some of the High Street and other areas. The city planners decided to replace the remains of severely damaged buildings (and many not very damaged ones) with 'modern' architecture. Given poor planning, architecture and lack of funds, but a need to re-build quickly - partly to be able to take advantage of such Government aid as there was - we ended up with the sort of buildings you refer to - such as http://www.exetermemories.co.uk/em/_streets/highstreet.php