Practically, any party who can contend, in the current context, has no real will to overturn the current context. It's simple political evolution, that there's no "phyletic gradualism" to this particular aspect of the electoral phenotype and the ability to shove over a sudden change ("punctuated equilibrium") is severely limited.
The closest it came to changing was with the LDs pushing for a change through their unanticipated coalition (analagous to a horizontal transfer of gene, or maybe better to suggest a temporary endosymbiotic analogue), but it arguably didn't take because, like many such random gene/function-hops that might occur when the usual machinery of cells gets a bit too friendly an external input, it just wasn't conveying the right level of advantage (and in the cell-line that last into the next generation of cells) to sustain itself.
(Outside of metaphore, it was arguably the wrong kind of non-FPTP, to make headway. By accident or design (some would say machination), the conceptual advantage attempted to be adopted wasn't even one that the majority of those who wanted change liked the idea of changing to. Under that time's national psychology, there was both small-c conservatism towards the idea of "sticking it to the current traditional system" and a too stoic "if we do change, I dislike this change we'd be changing to" by those who might otherwise have been the table-flippers who might have prefered to deal with the current bad hand by forcing the cards to be dealt again. Whoops, metaphore crept in again! Anyway, the problem was that the chosen alternative was too flawed to gain traction. Some say that better-qualified alternative-alternatives were ruled out just for this reason.)
Not that we're without alternative systems in the UK. Look to the devolved governments (or at least some parts of them, I'd have to check the less familiar ones). Though the system "designed never to have an overall majority" (Scotland) then got one anyway for quite a while (perhaps speaking volumes about the relative popularity of the party/perverseness of the electorate), and the one "designed for stable bilateral government" (Norn Ir'n) reveals itself easily stymied by one of the supposed partners throwing the toys out lf the shared pram at the slightest provocation (on top of other issues, just more general to the areas particular issues or politicians in general).
The 'way forward' is difficult, if you're thinking of juggling the House Of Commons methodology, because it's hard to establish significant, stable and beneficial changes to the established 'reproductive system'. You might luck on, two of those options, straight off the bat, but find it either not significant, not stable or not beneficial.
Better luck with the House Of Lords, maybe. I'm actually more towards the "it aint (too) broke, don't 'fix' it" camp there, myself, as I definitely don't want to have a carbon-copy "second elected house" (even offset, or given a slightly different term length, two 'populism-windblown' Houses removes what I'd term 'safeguards', even if others consider them to be 'blockers and spoilers'}, but I can see problems with the current setup. More problems, though, with the politically-upshoved life peers than with the hereditary ones (who are currently being whittled out, for the sake of 'democracy', the same democracy that still tries to stuff partisan statepeople onto the red benches).
Most of my own feelings about this aside, however, it might be easier/better to ease in some PR-representation into the HoL. Have a kind of 'party list' system whereby a number of issues are presented to the public. "How do you currently rank the issues of Education, Housing, Defence, Agriculture, Transport, [etc]" is the 'question', and the public answer determines how many 'new peers' are appointed from a queue of experts in said fields (if party-politicians must be involved, the list may be populated by them, but all parties get (proportional?) nomination-rights and the positions in the queues are interleved in some 'fair', and/or coin-toss, way). Thus we generate an influx of 'People's Peers', sufficiently different from 'Political Peers', which I would keep, but rate-limit.
As I would keep a Hereditary element. I feel that the less we rely upon ambitious individuals who would do anything to get into power, and the more we have who grow 'into service' more organically, the better. If there's any better solution against sheer megalomania flooding (or at least seeping) into both houses, it'd be to choose random newborns as proto-Lordships, let them grow up and attain a given age and then (for those not disqualified by something like criminality, emigration, having a particularly prominent occupation (which it is better for them to continue) or becoming an actual politician (...you can make your own joke about why these are excluded) shanghai them into being a Neo-Hereditary Lord. "Accident of birth"? That's an 'accident of birth'. Of course, they lack the "growing up, knowing that they'll likely be a Lord" nurturing which I think probably helps moderate the sheer ambition that we can't escape those who luck themselves into power through all the other means of achieving any sort of parliamentary status. What we want, though, are a mix of not-too-unwilling (but not ever actually eager candidates who aren't worried about comparative blips of public opinion and aren't likely to bet big upon holding onto power but know that they'll likely be having to juggle the results of their past choices a decade or several down the line, not "politic fast, retire young, go get jobs on some company boards off the back of knowing (and heing known by) the right people.
Anyway, you could conceivably tweak all that into the Lords, if done right, far easier than rebuilding the engine of the very same car as you're currently driving. (Not 'easily', but 'easier'.) Of course, 'future history' may prove me wrong, but it's more likely to change system through some actual (social?) revolution, i.e. waiting until the car spins off the road and taking the opportunity to tweak the motor before whoever next gets the wheel revs it all up again.