There are very, very few texts that discuss Britain's reaction to Irish independence in any depth. Most books on the subject, obviously, focus on Ireland and the Irish, with Britain only playing a role as the enemy. However, Britain's reaction was at several levels. Politically, in the sense of Party Politics, the Tories had always been opposed to Irish autonomy, and remained so. Indeed, the official Party name remained "The Conservative and Unionist Party" for many years. The Liberals, and even more so the relatively new Labour Party were very much in favour of Irish independence, as were the bulk of the British population, in as much as they cared. Rather like the civil population when foreign adventures are being undertaken, most people, whilst supporting "our boys" morally, oppose the adventure. Britain was much the same; post WW1, and the war- weariness engendered by it, the civil population wanted peace and they wanted their men to come home, so, a peace settlement in Ireland was welcomed by the troops, if not the senior officers, and was welcomed by the British population at large.

As /u/Second_Mate points out, the Treaty was deeply unpopular with many Conservatives/Unionists, with whom Lloyd George was in coalition (and on whom he depended for office after the 1918 election), and it's been widely cited as a reason for Conservative backbenchers bringing down the government in October 1922. There were other reasons: an ongoing economic slump which had fuelled demands for massive public spending cuts, and crisis with Turkey in September, which this time saw Conservatives reluctant to risk war. Ireland was a factor, but not the only one, and the Treaty stayed in place under the new Conservative government.