The attitude of the two main parties to those further to the right (for the Conservatives) or the left (for Labour) is very different. In the case of the Conservatives since Cameron, until very recently at least, the best word to use would be appeasement. We left the EU as a result. The attitude of Labour leaders (with the obvious exception of when Corbyn was leader) can be characterised as exclusion.
Is this asymmetry just the result of the choices of particular party leaders, or is there something more structural behind this. Another way of asking the same question is whether the success of UKIP (in terms of winning votes and influencing policy) could ever be duplicated on the left? Equally will Sunak suffer if he continues to marginalise the Brexit ultras, and would he have the political capital to replace his current home secretary, if he ever wished to do so (which he shows no sign of as yet) ?
In this way the Conservative led government of 2010-15 precluded the option of exclusion towards UKIP, so appeasement (and in particular conceding a referendum) was the only option left. As both UKIP and the Conservatives stressed the importance of controlling immigration, and the right wing press did all it could to worry voters about this issue, it was not credible of Cameron to portray UKIP’s stance on this as extreme. Yet Cameron was not prepared to damage the economy in order to meet his immigration targets, and did not want to leave the EU. In addition he made many potential UKIP voters worse off through austerity, yet blamed that on immigration. Thus he handed UKIP all the ammunition they needed to attract socially conservative voters. 
If this interpretation is correct, then two things might seem to follow. First, after the referendum result a form of Brexit where we left the Single Market was inevitable under a Conservative government. Anything otherwise would enhance Farage’s threat to the Conservatives. Second, even if Remain had narrowly won rather than lost the referendum Cameron would not have been able park the Brexit issue for long. His only hope of doing so would have been to start declaring the virtues of immigration, but the right wing press would not have played ball.
Could something similar happen with a Labour government and a more left wing political party? The Green party is associated in voters’ minds with one clear objective (tackling climate change and improving the environment more generally) just as UKIP was, and it tends to be more left wing than Labour (just as UKIP was more socially Conservative than the Conservatives). Yet while Labour is in opposition, like UKIP it does much better in local elections than in general elections.
Under a Labour government could the Greens repeat the success of UKIP? A little piece of imagination might help answer this. One of the centrepieces of Labour policy is increased public investment in greening the economy, which they argue is critical to both tackling climate change and improving economic prospects. Most Labour voters and members share that view. Suppose in government Labour continued to talk this talk, but at the same time completely failed to raise public investment in greening the economy because of deficit targets, say. The Guardian runs an article on how Starmer told the cabinet that it needs to “cut the Green crap” to meet those targets, and the economy continues to languish. In those circumstances it is not impossible to imagine the Green party becoming a vehicle of protest for Labour voters. .
The point of this bit of (hopefully) make believe is that the success or otherwise of a UKIP of the left depends rather more on what a Labour government does than what any more left wing party does. Cameron reneged on his pledges to control immigration, while still blaming immigration for the country’s economic woes, many of which were of his own making. Under FPTP if, and perhaps only if, a future Labour or Conservative government makes a central part of their political campaigning an issue central to their voters but on which they have little intention to act, will a party from the political flanks threaten the big two. 
 Tim Bale calls the right wing press ‘the party in the media’. While this phrase nicely captures the extent of its partisanship, it could be misread as implying an excessive degree of control by the political party. This media is ultimately controlled by its owners, and while on some occasions they may be happy for their papers to follow the party’s lead, on others it may be the party that has to bend to the tunes played by the press.
 This may affect both major parties. The non-partisan media tends to be both younger and so more socially liberal and also from privileged backgrounds resistant to radical left wing policies. This may cause difficulties for mainstream parties that push socially conservative policies or left wing economic policies. The former, however, may be neutralised through the influence of the press, or when third parties that are socially conservative have large scale popular support. Finally the BBC, through financial pressure and via senior appointments, has become less independent of the government.
 What Cameron and Osborne knew was that achieving a socially conservative goal (much lower immigration) would seriously harm the economy, and that this was the last thing they could afford to do when hitting the economy with austerity. The same was true for another policy favoured by social conservatives, Brexit, as the current government is finding out. Many of the conflicts/contradictions in the current government can be traced to this source.
 What is also possible is that, if the leader changes, the party membership may elect a more radical leader, much as Corbyn’s election represented frustration within the membership that the mainstream candidates seem to be accepting austerity just at the point that public dissatisfaction with the policy was growing.