Looking at the Brexit opinion polls it seems that the public is consistently in favor of staying in the EU ever since July 2017, with the gap between 'leave' and 'remain' slowly widening over time. So why do British politicians seem to ignore their electorate and keep pushing for leaving the EU instead of at least voting for a new referendum? Did any MPs mention the poll results in public discussions within the House of Commons?
It was discussed in the discussion on one of the online petitions. The standard Tory line against it is:
17.4 million people voted to leave. After that, 499 Members of Parliament voted in favour of invoking article 50, and 122 voted against
But Brexit, as currently being operated, is not a public-driven process. Even if it was, making it opinion-poll-driven on small daily fluctuations between 49/51 one way and the other would make no sense either.
No, the absolute driving motivation is to keep the Tory party together and in power. There has been a risk of a split over Europe since at least the days of John Major. Everyone is aware that under the FPTP system a split would be completely fatal to the party.
This causes an endless cycle of making concessions to one group within the party to prevent them defecting, followed by the discovery that those concessions have angered another wing of the party, or are infeasible to deliver, or the EU won't agree to them, and so on. It also explains the weird stasis where the government is unable to command a majority for its flagship legislation but has not yet lost a vote of no confidence.
There is no strong evidence that UK politicians are ignoring opinion polls. There is some evidence that the information in the opinion polls is more subtle than what's expressed in the headline figures. Consider this Survation poll for the Daily Mail, with fieldwork conducted on the 15th March 2019. The headline question is
Imagine there was a referendum tomorrow with the question. 'Should the United Kingdom remain a member of the European Union or leave the European Union?' How would you vote?
and for likely voters, ignoring don't knows, the breakdown is
in line with your question. However the tables also have the breakdown by party voted for the in the 2017 General Election. This gives a split for the Conservatives of 62% Leave - 38% Remain, of Labour 34% Leave, 66% Remain. Meanwhile the SNP and Liberal Democrats split towards Remain by 79% to 21% and 73% to 27% respectively.
So all told, the party in government is following the wishes of its electorate by attempting to deliver Brexit, even if it's at constant risk of sparking an internal political civil war about what Leaving actually means.
Meanwhile the Labour party is in an unfortunate position:
Any firm positive action in either direction will displease one wing or another of its party, leading to a perception of disunity which will cost it votes.
Many of those pro-Remain Labour supporters are in London and other large cities, whereas many battleground constituencies are pro-Leave or more evenly split. Hence a switch to Remain could cost it a disproportionate number of seats in the next election.
Just switching back to a pro-Remain stance allows soft attacks on being anti-democratic for ignoring the result of the 2016 referendum.
Meanwhile, the SNP have no real need to point to current opinion polls, since Scotland voted Remain and the party ran an anti-Brexit Manifesto. The last point is also true of the Liberal Democrats.
There's a lot more that could be said by looking at the figures comparing voting intentions now with voting patterns in 2016 (the short version is that relatively few people appear to be actively switching, so some of this signal is being driven by non-voters) but that would need a much larger meta-analysis of polls.
The answer is slightly different for each of the two main parties (Labour and Tories) but boils down to trying to upset as few people as possible with an eye on the next general election.
Consider the ramifications of changing their policy from delivering brexit to cancelling it. That would certainly annoy many millions of leave voters. On the other hand sticking with "we want a unicorn no-damage no-down-side brexit" and blaming the failure to deliver it on other people just plays into people's existing opinions that politicians are generally useless and you pick the least worst one.
The Tories have additional problems with any possible brexit ripping the party apart. That's why May tried for so long to not commit to anything, merely spewing literally meaningless slogans like "brexit means brexit".
Labour could more easily switch to remain, but a much better strategy for them is to support a confirmatory referendum. That way they can blame the failure to deliver on the Tories, and claim they delivered the will of the people with minimal responsibility. Of course some will blame them for even having a second referendum, but it's the least bad option for them.
In practical and on principle, being a slave to the polls is a bad idea.
The politicians who favour remaining in the EU ignored the polls from the 1970s to 2015 that have shown sometimes wide margins in favour of leaving. Arguing that something is right because a fickle public are currently in favour of it is politically risky, as it seems certain that the public mood will change and change again.
In principle, politicians are elected to lead. The principle of "Parliamentary sovereignty" is almost an article of dogma to many MPs. The idea that an MP will change their mind on a matter only because the opinion polls are against it. would erode this principle. In private and behind the façade of Westminster we know that they are very interested in opinion polls (and focus groups, and audience response surveys etc). But in public they try to act as if they are motivated only by their own judgement and understanding of an issue.
So why do British politicians seem to ignore their electorate and keep pushing for leaving the EU instead of at least voting for a new referendum
A GE where the lib Dems as the only 'remain' party did badly.
This leaves the implication of your 'at least' highly suspect; to ignore to votes for an opinion poll in order to do 'what the people want' is absurd. (There is an argument of ignoring the vote because of best interest, but that's another matter)
As to having another vote, that is a more complicated matter, but part of the criticism of the EU in the past was the 'neverendum', which almost certainly plays a part.
Also note that while arguing for a vote on the aspects of the deal can be sold as a practical matter, having a vote because of what opinion polls say is opening a whole can of worms. For instance, what if people are dissatisfied with a government halfway through a mandate?
I would imagine that its because that's not how democracy works.
In your standard democratic vote, everyone chooses, in good faith, a decision that they believe to be best. If then the votes result does not swing your way, you are unfortunately restricted by the democratic element of the vote to honour it anyways. Therefore, when the governing body sees a poll, of a small fraction of the voters, that requests the decision be revoked, they will ignore it. This is because it is not only undemocratic, but unfair.
Small polls or protests from a side that lost a vote is simply irrelevant. People may argue that opinions have changed, yet these small polls take such little proportions into account it'd be impossible to show anything without a second referendum. It's therefore necessary for the government to ignore minor protesting or polls and continue to deliver what people voted for, just as if a Prime minister was hated they would still serve their full term.
A second referendum would be undemocratic and seems as though the voters have played a coin flip, only to demand a second go when they lose. No matter what your stance on Brexit is, you have to respect the democracy of the situation. The government are not displaying ignorance of the voters, rather they have chosen to follow them, towards a decision that the government did not want.
The United Kingdom (UK) has geographic districts. Each Member of Parliament (MP) represents one small (compared to the country as a whole) geographic district. They are not elected by the country as a whole. As such, there is no reason for them to care about national opinion polls.
A better, but much more expensive, approach would be to run surveys in each district. However, that would increase costs, as accurate results still require larger sample sizes. There are 650 MPs. So even if polls could be a tenth the size for districts, that's still a considerable increase in the number of people polled.
I suspect, but of course cannot prove, that most Conservatives are from districts that still would vote Leave. As such, it is risky for them to vote Remain. This is one of the weaknesses of geographic districts. A comparatively small amount of voters in geographic districts can produce a legislative majority for a rather unpopular result.
There are three major areas that voted Remain: Northern Ireland; Scotland; London. Wales and that part of England outside London voted Leave. Conservatives have little representation in the areas that voted Remain, and what representation they do have may be in districts that voted Leave (even as the larger area voted Remain).
It's also worth noting that in the last general election, at least 82.4% of voters voted for a member of a party that had Leave in its manifesto. Because Labour (the second biggest party) and the Conservatives (first) both officially supported Leave. So it's not just the referendum.
TL;DR: national polling doesn't tell us voter preferences by district, which is what is important for political support.
Good answers already, but I'd like to add one more factor to consider - at this point, British politicians have good reason to doubt opinion polls.
In spring 2017 Theresa May and the Tories were way ahead of Labour in the polls, and believed to be in a position to win a much larger majority. She called a General Election based on the polls and lost badly to the point of needing the DUP's support to retain power.
Prior to the Brexit referendum, the polls were predicting a Remain victory, and you know how that turned out.
Why would they want to act based on a very close poll, when they have experience that seems a good reason to doubt those figures?
There are already a number of good answers given, but another view on it would simply be:
Because they are not asked the same question.
The polls opposes the remain and the leave, clear, binary choice. For the MPs and the government, the choices are much more fuzzy (in the mathematical sense of the word). They can fully leave (hard brexit), fully remain (no brexit), or partically leave. And that partially cover the whole range: from almost no connection, to leaving the institution but keeping all the rest in place.
If polls were to provide the whole range, the difference would not be that clear. But at the same time, many (most?) voters would be at a loss for a fully educated choice (understand all the consequences and implications).