The only possible end to bipartisan politics in the UK is the end of FPTP. See : USA.
Or Canada. Oh, wait...
FPTP + Polarized political culture. Canada is not as polarized as the US or UK (regionalism is too strong).
Ok. A little lecture on political science is in order, methinks
1. Duverger's law (DL). An empirical regularity that says that in places with FPTP electoral systems there will be
at most two major political forces, at least "in a long run". Originally formulated based on the experience of, primarily US and UK, though back at the time it seemed to describe Canada pretty well as well. In fact, even India did not do too badly for a long time (that is where the
at most part was relevant). The things changed.
2. Canada. That strange country that has had pretty much a stable 4-party system for quite some time, being a superficial embarassment of the DL. One of those parties is regional, of course, but 3 other parties compete in multiple regions.
3. India. A most strange FPTP democracy where a government coalition, on occasion include some 20 parties, with another 20 in opposition.
4. Clarification of Duverger's Law: it only applies election by election, and each parliamentary constituency is a separate election for such purposes. The idea here is that there may only be two serious candidates in a particular district, but that the party pairings might vary district by district. There is a further exception postulated: if it is not clear who are the two front-runners, one might get a 3-way split, but this is an unstable set of affairs. So, either the third guy is very close to the second, or very far behind - with anything inbetween being unlikely.
5. At this point, the general belief is that US is a two-party country because in addition to the FPTP for congress it has also the winner-take-all executive presidency (not quite FPTP, but close enough). So, DL applies nationwide and district by district, creating a particularly strong Duvergerian force. The reason I call it a "belief" is that US is pretty much the only such country, which is both presidential and FPTP, so proper testing of this belief is impossible.
6. There is a somewhat mixed, but, generally favorable, evidence for the district-by-district DL elsewhere. So, perhaps, we should take it as a decent description of reality. But the nationwide version is, obviously, not true.
7. Of course, this implies that the past outcomes in the UK, Canada, etc. were more of an artifact of a fairly accidental division of nations into two political camps, both uniformly present nationwide: Conservatives and Liberals (or, later, Labour in the UK). Notably, this division is not coming from DL - it is a completely different story. At some point for, whatever reason, Canada lost that geographic uniformity, replacing it with multiple regional bipartidisms. Note, that there is no theory here why it was one way, and why it changed.
8. Britain for a long time seemed to retain the nationwide bipartidism, though in recent years it has been increasingly restricted to England. One could dismiss a recent surge of LibDems as a "short-term"anomaly". Perhaps.
9. The problem is that we still have no explanation for that geographic uniformity. It is not implied by DL. Claiming it is "polarized culture" just gives it a name, but does not explain it. Nor does it, really, allow for predictions.
10. There is nothing in the systemic structure of British politics that forces geographically uniform bipartidism - Scotland and Wales illustrate this pretty well. Claiming it is there forever is not really based on any solid reasoning.
11. Furthermore, every bipartidism (local or national) is not enshrined in politics for enternity. If new issues arise, an old party may die, a new one may be born. Labs replacing Libs in the UK is a great example. There can be also temporary upheaval where such a replacement attempt is abortive: if it is not clear, who is the second and who is the third, DL logic suggests possible short-term multipartidism.
12. One thing that might promote such changes is an emergence of a new set of issues, or some other major shock to the political system. But, of course, England has not had any major shocks recently. And, in any case, it is simply polarized... Sorry, I have just stopped being fully serious.