I don't think we can think about the likelihood of the UK breaking up at this juncture; however, we can more or less see the contours of how that scenario might come about - and, like many answers to similar questions, it will depend on the shape of Brexit. Voters in Northern Ireland and Scotland voted in large majorities to remain in the EU in June's referendum unlike England and Wales. How a break-up of the UK might come about stems from different possible scenarios in each constituent nation of the UK.
In the case of Scotland a referendum on Scottish secession from the UK was held in 2015 and narrowly won by the unionists in large part thanks to the support the UK government received from EU member-states (such as Spain) that insisted an independent Scotland could not join the EU or by officials from the EU itself who suggested that if Scotland were to join it would have to negotiate its way in first. The Scottish National Party (SNP) had been campaigning that an independent Scotland would be able to somehow transfer the UK's EU status to itself after secession. Now the SNP is arguing that the Scottish voted to remain in the UK with the promise from the UK that the whole union would remain part of the EU and if that promise cannot be kept (primarily because of English nationalist sentiment that had, in part, been stirred up by the Scottish referendum) then Scotland should be permitted to hold a second independence referendum. At present there seems to be little support for a second referendum in Scotland but this could change dramatically if the UK government, under Prime Minister Theresa May, insists on the so-called "hard Brexit" of strict border controls with the EU. For example, if we take "freedom of movement", and recall that in most things in international relations, reciprocity will apply and UK citizens would, in that scenario, encounter much of the same restrictions imposed on EU nationals entering the UK themselves when trying to travel to the EU. A "soft Brexit", which would more or less be the same as the current situation but without UK representation at any EU institutions, would probably render the SNP's rallying cry moot.
Of course, and this issue has not been raised in the press (or to the best of my knowledge in the Academy) the UK government could ignore the SNP's threats, go for "hard Brexit" and refuse to recognise any Scottish right to a second referendum. What the consequences of such a hard-line approach to domestic politics might be are very hard to predict but violence seems extremely unlikely. Sadly, the same cannot be said for Northern Ireland.
Northern Ireland has largely been at peace and there has been much progress towards reconciliation between the Irish Catholic and Ulster Protestant communities there since the 1998 Good Friday Agreement. A key component of the peace agreement was the incorporation of the European Convention on Human Rights (ECHR) into UK law through the Human Rights Act of 1998 that was passed by an Act of Parliament. Another component of the peace agreement was the dismantling of the border crossings between the Republic of Ireland and Northern Ireland so that it came to resemble the land borders of all other EU member states. Finally, the EU has provided important investment into Northern Ireland institution building, infrastructure and the regional economy. Leaving the EU, whether it be "soft" or "hard", will mean that Northern Ireland will be taken out of many of the programmes that saw EU monies spent there but that in and of itself would probably not dangerously destabilise the region. The danger comes from a "hard Brexit", which would see some kind of border controls come into effect to control and monitor the flow of goods. This was seen in the EU's eastward expansion that brought in border controls to places that previously there had been none or were light and had some deleterious consequences (organised crime and wrecked borderland local economies) for inhabitants near those borders. For Northern Ireland, however, the consequences of bringing in border controls would be to deal a psychological blow to the Irish Catholic community for whom border controls with the Republic of Ireland were a manifestation of their enforced separation from Eire and London's control over their lives. An added danger comes if Prime Minister May is serious about her, let's call it, "hard Brexit-plus", of the UK not only leaving the single market and abandoning freedom of movement but also of withdrawing from the ECHR. This is an issue she has been consistent on since her time as Secretary of State for the Home Office and it is an institutional framework that has been demonised in the same sections of the English press that pushed for Brexit.
A hard border with the Republic of Ireland and withdrawal from the ECHR are, according to the Sein Fein republican party (and political arm of the Irish Republican Army (IRA)), red lines that would terminate the Good Friday Agreements. The end of peace and resumption of organised violence would not necessarily spell the end of Northern Ireland's continued membership in the UK but it would gravely imperil the stability of the entire system. Indeed, it would impose additional costs to a "hard Brexit" in terms of the added disruption to the economy there, the need to divert a British Army that has been cut to its minimal functioning ability there and the cuts to other services that would be required to fund ongoing military operations there in a bid to try to restore order. Under the current febrile climate that has been fomented by sections of the press and the Tory party there may be English nationalists that would welcome such a prospect. Indeed, and to all our dismay, they will find some supporters for that position among the more hard-line Ulster Unionists who were opposed to the Good Friday Agreement but no longer had the support to continue fighting.
This last part is also very important but there is really not enough data available to know what will happen and this concerns the strength of English nationalism expressed as anti-European, anti-Scottish, anti-Irish, anti-immigrant and stridently chauvinistic in character. Whether it is being amplified by the tabloid press and "political entrepreneurs" but is still marginal or if it has a much wider appeal than thought will come to light one way or another in the coming months and year. If Labour can create a blocking majority with the SNP, Liberal Democrats and others in the House of Commons then it might be possible to prevent the break-up of the UK and the turn towards this form of English nationalism. It will also require that coalition to deal with the ownership structure of the media in the UK but that is, of course, another question...