as the constitution doesn't have that high of a place in our identity it's more something for constitutional lawyers and higher courts. I think there's more tension between our constitutional law and that of the EU's law. for most people that's background noise I guess.
In this particular question there are also fundamental human rights that are part of the EU legislation where it is fully plausible that ECJ would render any law implementing the EU law. Like Ipred, where our politicians tried to pass laws following it but it got struck down again and again. No idea if they have given up yet.