British politics is now highly unresponsive to citizens. Three measures from the former colonies could greatly improve the situation: Canadian-style party funding, Australian-style party leadership elections, and American-style primary elections.
There ought to be close feedback loops between electors and politicians, but these have been gradually loosened and removed in the UK.
British legislators have to do what whips tell them, not what voters tell them. This explains some of those odd ways in which legislative outcomes differ offensively from public opinion.
Defying the whips gets an MP deselected, terminating his career.
But it gets worse: the whips are controlled by party leaders, who are not under the control of MPs or even voters.
The party leaders are "elected" by their "parties", and they can't be easily removed by MPs. The leaders are chosen by party members and, in the case of Labour, non-members (who need not even be eligible to vote in Parliamentary elections). The leaders don't have to satisfy opinion within their parliamentary parties because of this, and because the parties are not financially dependent on the membership, the leaders don't have to do what the party memberships want once elected, but what rich donors want.
This means the country is run for the benefit of the sixty-odd people rich enough to keep the parties afloat (millionaires in the case of the Tories, union bosses in the case of Labour, to the extent that the union bosses aren't millionaires themselves). Now this is much more pluralistic than, say, Russia or North Korea or France or wherever, but we could do a lot better. If the political system had to operate to the benefit of the sixty million citizens rather than the sixty millionaires, we'd doubtless be happier and more prosperous.
The way the Canadians have avoided this is by banning donations other than from individuals, and capping individual donations at $1000. Yes. A thousand bucks. We should do it here in Britain. Obviously this would bankrupt most of the major parties overnight, which would be no great loss.
Now parties which were financially accountable to their members would be a good start, but we can be stricter: Australian party leaders can be, and are, dispatched by a simple vote of the parliamentary party. (I believe Labor used to allow its National Executive Committee to dispatch the leader or dictate policy, and this was accorded unconstitutional). This means the party leadership cannot carry on policies opposed by substantial proportions of their own backbenchers, and these people are directly accountable to voters.
Those parliamentarians can be made more accountable to voters, by primary elections (and possibly by reforming how their pensions work - I don't know the details, but it seems that UK parliamentary pensions depended on whether you lost an election or did not contest it, and whether you contested it depended on ... you guessed it ...). There are different flavours of primary election, depending on who is allowed to vote, who is allowed to stand, and so on, but the details aren't so important. What matters is that voters rather than the party machine get to choose the candidate who goes forward to the general election.
Primary elections are a bit hard to engineer without fixed-term parliaments (which I oppose bitterly), and which, in practice, we won't always have. It can be done, though. UK parliamentary terms are five years. Primary elections ideally need to be some time before general elections, e.g., on year. Hung parliaments and other contingencies give rise to early elections, so one possible scheme is for a primary elections effectively to confer a status which lasts the length of one parliamenary term or until the next primary election, whichever finishes earlier. This is similar to how MP's actual terms in parliament work. Primaries could thus be scheduled for four years after the most recent general election, or five years after the most recent primary election, whichever comes earlier. This scheme can't be messed up by having, say, two general elections in a single year, as happened in the 1970s.
Primary elections, capped individual donations, and parliamentary control of party leadership are all such lightweight changes they could be imposed by political parties upon themselves, and don't risk destabilising other aspects of the constitution (unlike pretty much any political reform scheme you hear from the Rowntree Trust or the Lib Dems; one might be forgiven for thinking that was the point, some time).