It seems that the problem in this case is assuming that changing the build configuration profile will automatically update your Publish Profile to match - this is not the case.
If you have a single Publish Profile then you will have to manually change the "Settings" > "Publish" > "Configuration" setting each time you make a build, as this is what determine which config transform file is applied.
The recommended way to handle this is to create multiple Publish Profiles - one for each build configuration. For example, "Debug" and "Release", or in this case "Live" and "UAT". This way you can easily switch between the two profiles during the Publish process.
It is worth noting that there are other benefits to this approach than just managing your transforms. Having different profiles for different builds allows you to specify different targets too. For example, if you use the Web Deploy method you can send the application to different servers depending on your target environment (e.g. Testing or production)