I can't see us adopting a "big ideas" written constitution any time soon, but perhaps some of the structures may become more explicitly defined in legislation (in the way fundamental rights were enshrined in the Human Rights Act).
Who would we trust to reform our famously short written constitution? Ten words or thereabouts that was the fudge after the restoration, to give parliament sovereignty.
I certainly wouldn't want the modern political parties defining or improving this, yet who else? We probably need to lose a major war, so the victors can "give" us a reforming and benign constitution, like was successfully done a time or two in Europe.
The US constitution came from the revolution, and attempted to correct some of the issues of the time, but without needing a dictator. Interesting that the US president gets a broadly similar role and powers to George III, just with restrictions on length of office. We've reformed all of those powers away, they kept their 18th century monarch, oops president, and added a few new powers. :)
The baby steps often come from localised chaos - riot, civil disobedience and such. Don't think we could push for a good constitution that way. We'd get a biased, subservient to the parties one. But we'd be told we got what was agitated for.