I'm sure you know some of this but he was big friends with Thatcher and owns the Sun Newspaper. At peak it had 5 million daily readers and is still pretty big. In the 80s and 90s, this soft power helped move a section of the affluent working class away from Labour to support Thatcher's policies. Mixed of course with national jingoism and fearmongering. Blair famously met with Murdoch prior to 1997 because he felt he needed that alliance. The less said about Blair, the better.
More recently, his press has helped to stoke fears about immigration and a section of older, socially conservative working-class people voted Tory for the first time in 2019 to 'get Brexit done'. Our voter demographics now look typically American: Young middle-class people are left, older white working-class people vote right, though there are economic reasons for this also. This has become apparent only in the last five years.
Murdoch press is one of only a number of right-wing voices because our print media generally is heavily Conservative. Our print media also does a lot to set the agenda and the political centre of TV news. Because our broadcast media is more heavily regulated and because UK Sky has a broad-church marketing strategy, our version of Sky News is nowhere near as partisan as the Australian version. Conversely, the BBC is much more Conservative than a lot of people realise because its constantly in fear of losing the licence fee.
Arguably Murdoch's influence here is much less dominant than twenty years ago but it still exists. Sadly, UK media leans to the right anyway, irrespective of Murdoch.