Press

"Freedom of the press in Britain means freedom to print such of the proprietor's prejudices that the advertisers don't object to"

Americans like to believe their press is neutral. In the UK, hardly anyone bothers to maintain this pretence, and the press are openly engaged as political factions. The press circulation is much larger than the membership of the political parties, and has an impact beyond its actual numbers in terms of influencing opinion.

The three largest papers are the Sun (owned by Murdoch's News Corp), the Daily Mail (owned by Viscount Rothermere), and the Times (News Corp again). All have a right-wing anti-Europe stance, and will happily print inaccurate articles about Europe. Boris Johnson got a particularly bad reputation for this while working as a journalist before he was a minister. He is still employed as a journalist, and is in fact paid more for that than his role as a Minister of the Crown.

The press benefit from controversy: it sells papers. Readers enjoy outrage bait that panders to their prejudices; a pre-internet version of the "fake news" problem.

Sidebar: the Spiked/LM axis

There is a small group of people who used to write for Living Marxism until it was sued out of existence for denying the Bosnian genocide, then formed a successor Spiked!. They seem to be surprisingly influential in the commentariat despite their small size, and despite the intellectual incoherency of having gone straight from revolutionary communism to libertarianism without passing through common sense on the way.

BBC Question Time

A similar story prevails here. What ought to be a discussion show has succumbed to the temptation of getting on the most extreme guests possible, provoking highly strung arguments, and letting planted audience members inject outrageous questions.

Edit: another example of the revolving door: the man who ran the BBC's political output during the Brexit campaign was later appointed as Theresa May's director of communications and is a "hard Brexiteer".

Culture: The Establishment

The leaders of the country overwhelmingly come from a very narrow educational background: private school followed by Oxford PPE. This produces people with a bulletproof opinion of their own correctness and practice in intellectual bullying.

Culture: Ruins of the Empire

This mostly manifests as a belief that Britain can "punch above its weight", and therefore does not need to engage in international compromise. While the influence of this in negotiations with Europe is obvious, I would argue that it also prevails internally. (This is easily a graduate-thesis size investigation!). It seems to me that the country is run centrally as a tiny empire. Councils have little power and little funding autonomy; council tax rates can be capped centrally, and they are dependent on "block grant". The relatively recent devolved assemblies also get little respect from Westminster; the Scotland Office and Wales Office still exist despite seeming redundancy. The Scotland Office has a substantial budget for anti-independence campaigning.

The situation is even worse in Northern Ireland, land of No Surrender, where there has been no government for about two years following its collapse over a fraud scandal. Compromise is even more foreign there. People tend to forget that the UK had a live-fire civil war in living memory, but I think that matters to the uncompromisingness.

Systems: First Past the Post

The voting system at Westminster discourages coalition or minority governments, so there is no tradition of sound coalition-forming by seeking consensus.

Edit: as a piece of metacommentary, this is the first time I've noticed an answer getting significant numbers of upvotes and downvotes. I am aware that it can be considered opinionated, but I think that's a symptom of the polarisation that leads to no consensus. Politically engaged people disagree on increasingly basic things.

Most Prime Ministers don't need to seek consensus. The first past the post election system tends to produce parliaments with a clear majority for one party, the Prime Minister automatically the leader of the Majority Party. And since ministers are chosen from among MPs, they are mostly willing to vote the party line.

As most PMs don't need to seek consensus, to do so indicates that the PM is in a weakened state, in which she does not have a clear majority, and cannot get her own MPs to follow her directives in voting.

From a partisan point of view, doing a deal with the opposition is a form of betrayal, and this is why members of her party have criticised her for attempting to deal with Corbyn.

There is a further, more philosophical point: If the major parties agree on a consensus position, then there is no effective scrutiny or an real choice for the electorate. The opposition opposes a large amount of government business. This ensure that the acts of the government are closely examined by its opponents. In the situation in which a government forms a grand coalition is the parties reach a consensus, and there is nobody to tell them that it is wrong. When the election comes round, the parties either have to reject policies that they agreed to as part of the coalition (and appear hypocritical) or enter the election with very similar manifesto to their opponents.

We did have consensus politics throughout much of the 20th century, despite enormous social changes and strains. There were consensus-based governments from 1914 under Lloyd-George and Asquith, then after the 1931 depression a 'national government' of all parties was formed and 'national' governments continued until WWII. In 1940, another consensus-based coalition was led by Winston Churchill with other parties.

Although Attlee (Labour) won in 1945, from that time onwards was a period of 'national consensus'. This meant that whoever won, Conservative or Labour - there was a broad agreement on certain issues like housing, healthcare, social security etc. This post-war consensus was ended by Thatcher in 1979 who had a different, harder ideological form of Conservatism to her predecessors. Now, a group of very vocal pro-Thatcherite extremists mostly aligned to groups like the ERG and Legatum are making any form of compromise or consensus impossible.

The problem is May called an election in 2017 to get a bigger majority and a mandate for a harder Brexit, but she didn't get one from the voters. Instead of realising she had no mandate and reaching out to find a consensus in Parliament for Brexit, she tried too hard to placate the extremists in her party and gave money to the Northern Irish DUP in return for support. She also put 'red lines' through the kind of Brexit that was discussed in the referendum and around anything that both sides could form a consensus around. This has led to two diametrically opposed sides to Brexit and years wasted in negotiations with the EU which ONLY cover the terms of leaving.

The divisions in Parliament are really indicative of the divisions in the country, neither side will compromise, each side is furious at the other and any sensible compromise route cannot find a workable majority.

In addition to the other great answers, it's usual for politicians to continually denigrate their opposition.

Corbyn has been called a Marxist (!) and "dangerous", and that's just the milder stuff. The Tory party and Tory press have spent years demonizing him, so to cooperate with him now looks like doing a deal with the devil. They can't pivot fast enough.

Another worry is that Corbyn wants to be PM and will use the situation to further that goal.

There is also great suspicion from the Labour side that it's a trap, designed to spread blame for Brexit to the Labour party.

Yet in Britain, a government based on consensus appears controversial.

This statement draws far too broad a conclusion from a single anecdote. There are myriad issues upon which consensus decisions are made every year in Britain. But, usually these are on issues upon which there was a consensus in the first place, or upon which nobody had strong positions staked out in the most recent round of elections.

If anything, there is consensus on a broader array of issues in Britain both now and historically, than, for example, in the U.S. or France. For example, from 1931-1940, the U.K. was run by an all party coalition known as the National Government, to address a clear and present crisis (while the U.S. merely ended up with partisan dominant party rule by the Democratic Party in the Great Depression and Weimar Germany started out with a greatly divided partisan mix in parliament).

What makes this particular issue different is that it has a history of being highly partisan and divisive. Brexit split the U.K. almost 50-50 in the last referendum. It was a close vote and the political parties in parliament were divided on the issue (in addition to intraparty conflict on the issue).

Therefore, seeking out a broad coalition implicitly means that a lot of the people in the coalition have reversed themselves with respect to prior promises made to get them elected on that issue. This comes across as something of a breach of trust and so it is controversial.

Taking a point out of your question: Mind that in the German system h need for compromise not only comes from the recent tendency on "big coalitions" between CDU/CSU and SPD in the Bundestag but also from the second chamber, the Bundesrat.

In the Bundesrat each state's government has votes which they can only use together (i.e. if a state is having a coalition government they can't split the votes)

For many bills agreement by the Bundesrat is needed, thus even if a party had majority in the Bundestag they still need to cooperate with the majorities in the state governments.

A lot of good information in the other answers, but specifically on the Brexit issue, it should be pointed out that it is almost impossible to compromise between "in" and "out" as it is basically a binary choice. Any deal which involves following some of the EU rules but with no say in the making of those rules - making the U.K. into a "vassal state" - is inherently unsatisfactory for a large segment of both leave and remain voters.

This poll in "The Independent" provides evidence that in a new referendum Remain would beat Theresa May's deal by 61% to 39%, while if the choice was Remain versus No Deal then it would be a rather closer contest (57% to 43%). (Note, "The Independent" is a strong advocate of a second referendum. Recent polls from other sources show Remain having a much narrower lead.)

Consensus isn't controversial.

The complaints about Theresa May meeting with Jeremy Corbyn are from people in their respective parties who think that their own party's position is right and the other party's position is wrong. To such a person, "consensus" means "doing something that's less right and more wrong."

In what country would it take nearly three years, before a government, that is not in agreement gets the idea to cooperate with the opposition on a vitally important issue? It is really incredible, that this is happening in [Great Britain] only now.”

This is a complete, and probably deliberately, untrue presentation of the facts. The negotiated agreement with the EU is not three years old, so it would have been a bit difficult for May to have negotiated a way to get her agreement passed beforehand.

As for his actual statement, I would have hoped that a Dutch politician would have heard of Belgium.
https://en.m.wikipedia.org/wiki/2007%E2%80%9311_Belgian_political_crisis as anyone reading the article will note, there were a couple of high-profile EU politicians involved in their record 500- odd says without a proper government, so perhaps Timmermans can ask them to give the UK a lecture on consensus politics