Unlike what many people think, the UK has a written constitution in the form of various laws. However, Parliament (more specifically, the House of Commons) can in theory alter any of these laws at any time. Such is likely only in case it would further solidify Parliament's power and preeminence. Brexit is the last time it happened, when the perceived subordination to the EU was removed.

It's admittedly a messy state of affairs and many important processes are indeed defined and motivated by convention and tradition only. However, many of these traditions came about whenever the British were forced to deal with events or a constitutional crisis. For example, the monarch does not enter the Commons and Black Rod is admitted only after striking three times exactly because of such events.

January 6th is doomed in the UK for the same reasons it was doomed to fail in the USA: it was based on a flawed understanding how the government really works. The British would have doubtlessly dispensed with tradition and taken expedient measures, and those would become the new Tradition, regardless of whether it is subsequently embossed in Law or not.