Hello,
Welcome to the forum. Your contract is with Bett homes you are not party to the contract that Bett has with Glenfield plumbers. They are trying to fob you off.
New Homes do not have the protections of the Sale of Goods Act 1979, or the Supply of Goods and Services Act 1982. This means you have more rights buying an electric blanked from Argos. Before these acts came into force consumers had to rely on common law and indeed this still applies to new build homes and would require the installation to be completed in a "workmanlike" manner - i.e. installed correctly.
If the claim is less than £10,000 you can start a small claim for £35 using
Moneyclaim online - slightly different process in Scotland.
Given them one last chance to fix it stating that you will have no choice to start legal proceedings within 14 days if the problem is not rectified. And mention that under common law it is reasonable to expect the heating to be installed in a workmanlike manner. Plus, reiterate that your contract is with Bett not the plumbers.
Hope this helps
Tony