Have Britain not conquer India, Australia or Southeast Asia and instead by the end of the Napoleonic wars rule those Islands along with their other Mediterranean possessions. Have the British Tories such as Pitt and Liverpooll go in for giving the aristocracy of these islands representation in parliament and have all their OTL and ATL Meditteranian possessions be integral parts of the UK, like Ireland or Wales. British laws applying to the United Kingdom apply here as much as in the British Isles. Have Catholic Emancipation in 1829 allow the elites to become MPs and have the subsequent reform acts enfranchise more and more people.
I personally see it benefitting Britain and British Politics hugely. The British colonies would obviously become more accustomed to and influenced by people and customs that aren't Anglo-Celtic. I also see them being inspired by Parnellite constitutional Nationalism in Ireland and teaming up with the Irish so that by 1914, the Redmondite Hiberno-Sardinio etc Parliamentary Party holds the balance of power and the UK becomes federal.