What you legally could do is to start a petition for direct democracy according to the Swiss model, and then get enough people to sign it to then throw this huge petition at the feet of government and argue it should become a binding referendum and see how they react.
True legitimacy comes from the people, and if the people are by vast majority in agreement that they should have a different form of democracy it should be legitimate to abolish the previous one and replace it as long as the system is also within the bounds of human rights and protective enough of the rights of minority groups. That is to say, it is within the philosophy of self determination that this right could not be denied to the people by a class of careerist politicians.
It is a route to a peacefull transformation that can act as a strong vessel for the people to "take their government back". and there is "potential", if the British people can prove smart enough to rule themselves in good order. The Swiss have this empowering form of democracy, and rest assure its not a perfect system, but they arnt doing all that bad for themselves to be fair. It is a system that works somewhat better for the lower and middle class regardless that it isnt a Utopia, because these classes have those politicians more on a short leech. Then again the people then need to take responsibility for their choices, invariably building some tradition and culture of public consciousness and civil responsabillety that can arguably only start when the people also "learn by doing".
Yes its populist, very much so. The people dont get to blame it on the politicians anymore when its self rule, and neither can populist politicians. It takes a lot of sail out of opportunist poppulists if they cant direct their arrows at "some evol government" when it proves that the people ARE the govermetnment, because that would only spawn a reaction of "well F u who you think you are lets get your poppularity rekt".
It kinda love it though, especially if the poppulous would succeed with it too ofcourse, that they can stick it like that to the collectiove class of carreerist politicians and reduce their importance to a fraction of what it was in a dry fashion that has broad democratic support. Just show them the exit door, no guarantee that we wont reduce your pensions a bit either sorry about that.
Mind you, it comes at an ironic time, that just as their is a labor PM that is in power such apparent right wing elements come out of the woodwork to function as a destabilizing element for this "different direction", while apparently also seeming to blame this new govermnment for policies that ran for decades which they actually opposed from the opposition. The smell of this current violent process is quite in contrast with the ideal here put above, it is indeed completely anti-democratic in fashion and seems hostile to a aproach of finding utlillitarian compromises in a diverse enviroment, as a means to do well for all people.