This is a partial answer, concentrating on the calls for the reform of the Army, which is the portion of this I know the most about. The short answer is yes and no. The press coverage (led by William Howard Russell of The Times, arguably the world’s first special correspondent) definitely shaped public opinion. However, there is much less evidence that politicians or military officers paid significant attention to it. The press attention's major successes came after the war, when public opinion drove a number of reforming campaigns.

Journalists took full advantage of the new speed of communication technology to imbed themselves in the expeditionary forces in the Crimea, sending home reports that were eagerly read and reprinted across Western Europe. The readers were in for a surprise, however. Rather than more detailed depictions of the type of victories won by British forces in the Iberian Peninsula forty years earlier, they found the papers full of reports outlining poor conditions and incompetence. (See, for example: Special Correspondent from the Crimea, The Times, November 19, 1855) The press and the public immediately began to seek explanations (or scapegoats) for this incompetence. Lord Raglan, the commanding British General in the Crimea, was the first to come under fire. Raglan was a brave soldier who had served under Wellington in the Peninsula and had held varying political and administrative positions since, but his mismanagement of the British Army drove The Times’ special correspondent W. H. Russell to dismiss Raglan in a private letter to John Delane, then editor of The Times, as “utterly incompetent to lead an Army through any arduous task.” (W. H. Russell to John Delane, November 9, 1854, John Black Atkins, The Life of Sir William Howard Russell, C.V.O., LL.D. The First Special Correspondent (London: John Murray, 1911), I:173) Delane, convinced by Russell’s letters and other reports, brought the significant power of his paper to bear on Raglan, writing to Russell that he had “opened fire on Lord Raglan and the General Staff. According to all accounts, their incapacity has been most gross, and it is to that and to the supineness of the General that the terrible losses we have undergone are principally to be attributed.” (Delane to Russell, January 4, 1855, Atkins, Life of Russell, I:186) His writers were barely more restrained, informing their public that “the noblest Army ever sent from these shores has been sacrificed to the grossest mismanagement.” (Leader, The Times, December 23, 1854) Before popular opinion could rise to the level of forcing the government’s hand and replacing him, Raglan died of a combination of dysentery and depression.

With Raglan gone, The Times turned their attention to a wider target, and one, perhaps, more deserving than the late general: the British officer corps. While the nearly forty years of peace between Waterloo and the Crimea had been limited to Europe, and portions of the British Army had been engaged in colonial fighting almost constantly in that time, there were elite regiments who, through a lack of overseas service, had come to resemble social clubs more than military organizations. Jacob Omnium, one of The Times’ senior correspondents, noted in October 1859 that “during the 40 year peace the British cavalry had an easy and pleasant time of it. Consisting of 26 regiments, with here and there a rare exception, none were ever called upon to serve in our colonies, and only 16 were liable to proceed, in their turn, to India.” (Jacob Omnium, Letter on the Purchase System, The Times, October 19, 1859) Officers in the Brigade of Guards, an elite ranked above the cavalry, took up to eight months of leave per year, and could live wherever they wanted. ( Anthony P. C. Bruce, The Purchase System in the British Army, 1660-1871 (London: Royal Historical Society, 1980), 51) Even The Economist, a publication firmly in support of the military status quo, had to admit that the peacetime life of Britain’s officers did not prepare them, mentally or physically, for the trials of war. In a December 1854 article celebrating the British soldier’s discipline and courage in the face of Russia’s Army and her winter, The Economist compared life in the trenches to the officer’s “previous life… in the lap of the most enervating luxury and indulgence.” These were gentlemen, The Economist continued, who were “accustomed to the indulgences of the London Clubs, to silver dressing cases, to the most careful and elaborate toilettes, who never washed without eau-de-Cologne or almond soap, who rejoiced in the spotless polish of their varnished boots, and who ‘gave their whole minds’ to the tie of their cravats.” (The Economist, December 16, 1854) This level of idle dandiness vanished immediately on campaign, where the officers had to endure the same privations as their men, but while the men were trapped there, Russell noted with disfavor the habit of officers to escape the tedium of camp life by returning home on leave to deal with “urgent private affairs.” (Atkins, Life of Russell, I:242n) His objection was not anything to do with implied cowardice, but rather that he felt (and here he showed more knowledge of the psychology of war and duty than the officers did), that their leadership responsibilities were just as pressing in camp as they were on the battlefield. (Atkins, Life of Russell, I:242n; The Leeds Mercury, November 15, 1855)

The Times, with its trailblazing war correspondent and a daily circulation of over 50 thousand copies, was a formidable mouthpiece for the reform movement, but more was needed. (W. T. Coggeshall, The Newspaper Record, Containing a Complete List of Newspapers and Periodicals in the United States, Canadas, and Great Britain, together with a sketch of the origins and process of printing, with some facts about newspapers in Europe and America (Philadelphia: Lay & Brother, 1856), 95) The Times may have dominated the daily press, but amongst the popular weekly press, it was the conservative pro-purchase publications that held sway. (The Times nearest rival in the dailies was the conservative Morning Post, with a circulation of 2667. In contrast, The Illustrated London News, The Economist, and John Bull, three of the leading pro-purchase weeklies, had a combined circulation of around 88 thousand. Coggeshall, Newspaper Record, 87-89) To counter the conservative weeklies, and to try and emulate the popular upswing in support created by the Anti-Corn Law League, the Army Reform Association (ARA) was founded in 1855 to “keep before the public the necessity for a thorough change of system.” (The Army Reformer, March 10, 1855) Despite its broad title, the ARA and its short-lived weekly newspaper The Army Reformer had one goal: “the total abolition of the present pernicious system of purchase by sale of commissions.” (Announcement of the formation of the Army Reform Association, Hampshire Advertiser & Salisbury Guardian, May 12, 1855)

The truly staggering amount of correspondence generated in the Crimea, and the newly-found immediacy of war reporting in the press, make the story of a nation snapped out of its stupor by the sudden shock of war a compelling one. But while there is evidence aplenty of the sudden shock, there is very little that suggests that awakening was anything but temporary, or that it was followed up by long-lasting action of any sort. The inaction of the government and the safe suggestions of the Commission, however radical they were to certain elements, demonstrate the inaccuracy of that school of historiography that puts the Crimea at the start of British military reform. To put it simply, the government and the Army realized they had to be seen to be responding to the popular media’s calls for reform, but had no real intention of reforming a system that had worked well for a long time and had, despite setbacks, emerged victorious in the Crimea. The main tool used for this placating deception was the 1857 Commission on the Purchase and Sale of Commissions in the Army, whose recommendations were then stymied by a coalition of objections based on cost, fear of stagnation, fear of corruption, and conservatism. True reform required either continual popular, majority political, or royal support. In the 1850s, the only threat came from a surge in popular interest caused by the war. If the military waited and delayed, the war’s end would cause the support to peter out, allowing things to continue in the status quo antebellum, which is precisely what happened.

The purchase system was finally abolished in 1871 by a combination of parliamentary procedure and Royal Warrant. (For the history of this strange compromise, see Bruce, The Purchase System, 128-143) Reform was not achieved until the government decided it was necessary. Although popular support was there (albeit of a more gradual, long-term nature and not the sudden explosion after the Crimea), it was not until the government decided that reform was necessary that anything was achieved.