Why does the UK have a two-party system?

While the combined vote share of the Labour and Conservative parties has fallen from 96.8% in 1951 to 67.3% in 2015, the percentage of seats held by the two parties has only dropped from 98.6% (1951) to 86.5% (2015).
This is because the electoral system used for general elections wastes so many votes and makes the distribution of a party’s support as important as the strength of its support.
In the 2015 General Election, UKIP received over 3.8 million votes, but won only a single seat. This is because these 3.8 million voters were quite widely dispersed over the 624 constituencies where the party fielded candidates.
UKIP came second in 120 constituencies, but, under FPTP, a third party could come second in every single constituency, and still not win a single seat. Without a more proportional electoral system, it will remain very difficult for third parties to gain power, because, even though the two main parties enjoy far weaker support than decades ago, it is still strong enough to give them a plurality of the vote in most constituencies.
The reason why national parties like the SNP have been able to break through in spite of FPTP, is that their support is concentrated in a particular region. The SNP won 56 seats in the 2015 General Election with only 1.45 million votes.
This is because the party had only fielded candidates in Scotland’s 59 constituencies. Their concentrated support gave them the plurality they needed in these constituencies.

The Conservative and Labour parties still dominate in many other English elections.
Despite the large number of candidates, and the use of the supplementary vote system, the Labour and Conservative candidates collectively received 79.2% of first preference votes in the 2016 London Mayoral Election.
The two parties were equally central to the 2016 London Assembly election, collectively receiving 74.5% of constituency votes, and even 69.5% of the more proportional London-wide votes, giving the two parties 80% of the available seats.
In the first Police and Crime Commissioner elections in 2012, independent candidates won 12 of the 40 contests, which was viewed by some as a sign of the decreasing influence of the two main parties. However, in 2016, with turnout much higher, and many voters going to the polls to vote in other elections, only three independent candidates were successful. Instead, Conservative and Labour candidates won 35 of the 40 PCC elections.
On the same day, the former independent Mayor of Bristol was also defeated by the Labour candidate. In the 2016 English Council Elections, the two main parties both lost seats, but this still left them with 78% of the seats that were contested.

Our main parties are adaptable enough to adopt third party policies, undermining their appeal.
UKIP gained its first elected MP in October 2014, when Douglas Carswell won a by-election in Clacton, triggered by his decision to resign after defecting from the Conservatives.
Many speculated that the party’s increasingly strong performance in by-elections signalled that they would make considerable gains in the 2015 General Election.
However, the Conservative Party’s promise to hold an in/out EU referendum may have undermined UKIP’s appeal.
As both the Conservative and Labour parties supported ‘Remain’ in the referendum, it was then speculated that the victory of the ‘Leave’ campaign could again bolster UKIP’s support in future by-elections.
However, to date, the party has continued to struggle. With the Conservatives now largely united behind ‘Brexit’, and with Theresa May adopting other policies long promoted by UKIP, such as an expansion of grammar schools, it is again much harder for UKIP to stake out a unique position.
The adaptability of the two main parties can mean that third parties can find that their most popular positions are co-opted, giving them influence, but not electoral success.