This answer does not reflect reality in the UK. Party discipline is
extremely important in the UK system, maintained by the "whips". Few UK MPs carry their own seat as an individual, although good constituency work can be worth a few thousand votes. One or two have managed to run as independents; Peter Law (ex-Labour protesting against all-women shortlists), Richard Taylor (Kidderminster Hospital).
Tony Blair was notorious for imposing ideological coherence from the top on the Labour party with pagers in exactly the way you are talking about: MPs were expected to keep exactly to the media line agreed by Blair and Campbell, and hide all dissent to avoid "PARTY SPLIT" headlines.
Note how the conservative party managed to campaign against itself for the Brexit referendum and has now seemingly zipped that right up again. Although the three Brexit ministers keep issuing daft announcements that have to be countermanded by the PM (this is called "cabinet collective irresponsibility").
It's worth comparing UK parliamentary elections with the regional parliament elections, which are all run on different systems with very different results.
US parties have traditionally been extremely strong. What has happened is that the US rightwing media "bubble" has taken over doctrine and thrown away the doublespeak. Trump is winning because he can say all the unpleasant things that other Republicans feel they have to dogwhistle or hint at.