I did a bit of testing on this and determined that you do not have the proper element names in the if statement. It looks like you are referencing the id instead of the name for most of these. When an element is posted it is posted under the name of the element, not the id. This is the trick to working with radio groups, if you have two radio buttons with the same name they will be in the same group. You need to ensure that they each have a unique id though.
When I update the if statement to work with the names of the elements it seems to work as expected, here is the updated if statement that I found to be working:
if((isset($_POST['House_Keeping_group_Council_Tax']) && $_POST['House_Keeping_group_Council_Tax'] == 1) || (isset($_POST['House_Keeping_group_TV_Licence']) && $_POST['House_Keeping_group_TV_Licence'] == 1) || (isset($_POST['House_Keeping_group_Water_Company']) && $_POST['House_Keeping_group_Water_Company'] == 1) || (isset($_POST['House_Keeping_group_Council_Tax_2']) && $_POST['House_Keeping_group_Council_Tax_2'] == 1) || (isset($_POST['House_Keeping_group_TV_Licence_2']) && $_POST['House_Keeping_group_TV_Licence_2'] == 1) || (isset($_POST['House_Keeping_group_Water_Company_2']) && $_POST['House_Keeping_group_Water_Company_2'] == 1)){
You should make sure to updates yours like this, also make sure that you have the else statement to set the alternative redirect page.