The way for them to achieve this, gromit, is to lobby their MPs and try to have a bill introduced into Parliament. Then all they have to do is to have it passed by the Commons and the Lords.
At the moment, of course, this is most unlikely to happen. Even legislation that an overwhelming majority of the electorate would like to see in place has no chance of enactment if it is not included in either of the main party�s wish lists.
On a more serious note, I see no reason why this code (I won�t call it law) should be given any credence in the UK. Muslims are still (at least for the present) still very much in the minority and, as good as it may or may not be �Sharia Law� has no place here. I also note in the articles to which you provide links that those concerned �...are interested only in integrating personal law�. Without a doubt, this would be the thin end of a very thick wedge and it would not be long before the less pleasant aspects of this code are being countenanced.
Furthermore, it has already been seen that conflicts of culture are already causing enormous problems for some (mainly young) Muslims here. To support further division by providing a separate framework to which only some people subscribe is a recipe for disaster. The operation of these so-called courts serves only to emphasise such divisions and, far from being encouraged, those running them should be heavily criticised.
This country has a mature and legal framework which, whilst not perfect, is a compromise with which most of the electorate feel comfortable. Those who dislike it so intensely should try their luck elsewhere.