The question is how we get out of the impasse in the immediate term, and what we then do over the medium term to recover from the trauma of recent brexit-and-austerity-dominated years.
Any answer to this question must possess a natural appeal, otherwise it will lead to years or decades of grumbling resentment.
Few such answers offer themselves. We are in a Gramsci world, never more so, in which (paraphrasing) monstrous spectres stalk our politics as the dying old order blocks the birth of the new.
Our polity needs a constitutional relaunch. Those who did not grasp this some decades ago must surely grasp it now in the light of the frustration which brought us brexit. The very worst outcome from the current brexit seizure would therefore be an unchanged constitutional landscape still dominated by dinosaur parties given artificially strong majorities by first-past-the-post, with an upper house still appointed by prime ministers (and, yes, 26 church appointments and 92 hereditaries). To move from this to something more effective and fair, the referendum we really did need was one seeking adoption of a new constitution (one designed by thinking people, concerned with the interests of humanity as a whole, as well as those of the people of all the British and Irish nations). But we are where we are.
So we have two imperatives: escape from the impasse, yes - but by moving into a new and better polity; not by going back to dodgy-business-as-usual with power returned to one or other of the two parties of government either of which can sustain itself in power through the support of special-interest groups such as the DUP or Liberal Democrats or, one day if Labour can stomach it, the SNP. This system is morally repugnant and bankrupt and it will never deliver a politics sufficiently attractive to the vast majority of the population.
Over the medium-to-long term, there seems little point to the union of the three nations plus one province. Certainly we would not want a situation in which Scotland, in particular, feels trapped against its will to remain in the UK by the sheer scale, risk and complexity of the exit process (as now amply demonstrated by the brexit catastrophe). That cannot be healthy; instead, we should agree UK-wide a gradual transition to ever-more-complete devolution with each nation given time to build its own institutions ready for the very likely event that we mutually agree to separate. By such a long, open and transparent bureaucracy-led transition, we avoid the dislocation and chaos which would follow something akin to a unilateral declaration of independence - a hard Scoxit - but we also avoid the alternative: the slow sepsis of a Scotland unable to escape, resentful, depressed, brooding and brittle.
There is an option which matches quite well each of these requirements.
An English Parliament must be offered to the people as a compensation for a postponed brexit. The sales pitch would be "the Westminster House of Commons has failed to resolve the brexit question; a chance to do so will now be offered to an English Parliament, after a suitable period of time has elapsed to allow the parliament to be established". That English Parliament must be elected by a reasonably proportional voting system - one, crucially, which obliges large parties to stand multiple candidates in each constituency in order to take candidate selection out of the hands of the fringe elements which monopolise the process at branch level under the current party-dominated system. The First-past-the-post-with-allocation (FPTP-A) system would be one such.
This one act replies to the demand from the English for their voices to be heard - far more so than does brexit.
The obstacles to brexit which have so frustrated brexiters - the inner-Irish border question; the large pro-EU vote in the referendum in Scotland, in the big English cities, and in Northern Ireland; the logjam in the House of Commons - are all at least partly addressed at the same time. Scotland would be much better able to choose to leave the UK whilst retaining EU membership; Northern Ireland would be able to apply to become a province of Scotland or of the Republic of Ireland thereby avoiding being dragged out of the EU by brexit; the proportional vote for the English Parliament would reassure both leavers and remainers that their voice had been heard and, if there were eventually a parliamentary vote in favour of leaving, it need not be seen at that point to have been a coup d'état by a smallish faction of the population and of the parliamentarians - in sharp contrast to what we have witnessed during the first attempt at brexit. This new brexit would need a majority of parliament to back specific proposals put forward by a majority of the cabinet. If, after due and informed debate, such a result occurred, brexit would have to happen but remainers would recognise that the debate had at least taken place.
The House of Commons, meanwhile, would be dissolved and the lords would meet in a temporary chamber thus allowing the palace of Westminster to be refurbished in a cost-effective way rather than the chaos which is otherwise going to accompany the ludicrous attempt to upgrade the buildings whilst they remain occupied by a working parliament over the next decade or longer. The English Parliament (assuming it has 533 representatives, one for each existing English constituency) would consider issues mostly by reading texts and voting online thus avoiding the need for a large, dedicated building. Moving away from the debating society traditions, in which a premium is placed on rhetorical skills, to a more rational system of reflection and online written comment, leading to an online vote, would be a significant improvement and would interest a wider range of people in becoming MPs. The lack of visibility inherent in this could be partly offset by staging regular public events attended by all MPs, to be held in a suitably large and securable venue (and preferably a different one each time, moving around England). The English Parliament on tour.
The other two nations, and the province of Northern Ireland, would be represented during this period by guaranteed seats (one or two each) in a UK-wide cabinet. These seats, it is assumed, would be occupied by the first minister of each nation or by representatives of each of the two largest parties by popular vote in each nation. Through this function, and through the presence of the devolved assemblies and parliament, Scotland Wales and Northern Ireland ought not to lay any charge of lack of representation.
To accompany proportional voting in England, robust rules would govern the appointment of English members to cabinet seats (in a process which would be overseen ideally by an elected head of state, but let's not be too ambitious just yet - in the remaining lifetime of monarchy, instead, a royal commission would be appointed by the privy council to attend cabinet meetings and to referee (and report to the public on) the political process). The rules on cabinet appointments would award seats in line with the share either of the popular vote or of the seats in the English parliament (given that the other nations and the province are guaranteed one or two seats each). Parties would therefore be obliged to cooperate. The commission would castigate very publicly any recalcitrant behaviour or skullduggery. Under this system, far less legislation would be passed and far less vandalistic change made to the organs of our state; indeed the only changes which would be passed would be those commanding a sufficiently large majority of the cabinet (and the threshold could be set at more than half for certain categories of decision). The House of Lords would be replaced by a senate, of around one hundred to two hundred members, with seats awarded by a royal commission to each significant stakeholder-group within our society: regions, charities, philosophies (atheists, secularists, humanists, and religions), sport, arts, media, business sectors, scientific research and ethics bodies etc. Each stakeholder-group would be responsible for choosing and nominating its allocation of members, subject to scrutiny by an appointments commission.
This would be a unique, custom-designed solution for Britain. If, once established, the English parliament voted to take England out of the EU, but with the Scottish parliament and Northern Ireland assembly still wanting to stay in, the preparations already made by that point in time (the processes of preparing the bureaucracy in each nation for a return to sovereignty) would allow for a mutual separation in which Scotland would become an independent nation within the EU; Northern Ireland could choose to apply to Scotland or the republic for protection; and Wales would likely choose to be a part of England pro-tem (which, legally, it already is) or alternatively independence within the EU.