"If the government that is formed after a hung parliament is to be in accord with the canons of democracy, the parties must tell the voters before rather than after the election, both with whom they would consider forming a coalition, or supporting by a confidence and supply agreement, and also which items in their manifestoes are negotiable and which are not." Vernon Bogdanor in his 2015 essay "The crisis of the constitution".
Our politics has moved far from the two-party dominance which peaked in the 1951 election but our constitution has not. Parties cannot be trusted to announce their coalition groupings and policy platforms, and to stick to them, in the way Bogdanor suggests. Instead, a referee of some form must apply constitutional laws which dictate the method to be used for forming a government. It must grant cabinet seats to the parties in line with their share of the popular vote (possibly with a guaranteed pair of seats for Scotland, and a pair for Wales, and a pair for Northern Ireland), until a majority of the votes cast is represented in cabinet. This would mean, for the foreseeable future, a government made up of MPs from the Conservative and Labour parties (plus those from the first and second parties in Scotland, Wales and Northern Ireland). We need balance, compromise, and stability; not the pendulum swinging from the vandalism of the right back to the vandalism of the left and back to the right every few years. Yes, this will mean constant squabbling and grand-standing but it will avoid the excesses enabled by our "elective dictatorship" such as the poll tax. Government will no longer be able to inflict drastic changes on our society with the backing of (often) less than forty percent of the electorate. Instead, we will only get changes which have at least tacit support from around sixty or seventy percent.
Next, how to elect the members to the House of Commons. I dare say the voters of England cannot be weaned off the "constituency link" (one single MP for each constituency of around sixth to seventy thousand voters, elected locally by a simple and clear method). This rejects STV, AMS and regional list systems. But the vital principle is that the parties should be represented in the House of Commons in line with their share of the popular vote. This is very easy to do. The process has to be done separately for each UK-nation (Wales, Scotland, Northern Ireland and England). Let's take England (and then it's the same process for the others). First, give seats to any candidates (including independents) who have won a local majority. Then for the "no-overall-majority constituencies" (NOMCs) start with the party which won the highest share of the popular vote in England. Rank the constituencies of England by the percentage of the vote won by that party (so the top-ranked constituency would be the one where the party achieved the highest share of the local votes cast). As an example, if the ABC party won thirty-five percent of the votes cast in England, it should be given 187 seats (35 percent of 533). Let's say it had already won 105 seats by simple local majorities, leaving 82 seats to be allocated. In this case, the top-ranked 82 NOMCs would go to the ABC party. (If the ABC party stood multiple candidates in any of those constituencies, the candidate in each case with the most votes is the one returned). Then do the same for the party with the second-highest vote count in England. And then the third ... etc. The same process happens in parallel for Wales and Scotland.
The only downside of this method is that constituencies with a wide and even spread of votes would get an MP from a party which did not win locally. This is where the biggest sell is needed. Voters need to get the message that overall this is the fairest way of electing the House of Commons; some constituencies will get the wrong MP but it's a price worth paying.