As the question asks, I am curious as to how British opinion (both government and popular) of their own naval status was challenged by the Japanese victory at Tsushima. As far as I know, Tsushima was the first major naval battle involving modern battleships and the first naval battle since Trafalgar that was overwhelmingly decisive for the winning side. The British launched the Dreadnought class battleships only a year later, so I am wondering how connected these events were.
Tsushima did not challenge Britain's feelings of naval superiority. There were three main reasons for this. Firstly, the Royal Navy greatly outnumbered any other of the world's navies. The Naval Defence Act of 1889 put the RN on the Two Power Standard - requiring it to maintain a strength equivalent to the next two strongest powers in the world. While this wasn't entirely followed, the RN was still considerably stronger than any other navy in 1905. It had twenty-eight battleships in its active strength, plus another thirteen in its reserves. This compared to seven for the IJN, or twenty-three total for the Russian Navy. Secondly, Britain had been in a defensive alliance with Japan since 1902. The Anglo-Japanese Alliance committed Japan to coming to Britain's aid in the event of Britain becoming embroiled in a war with two or more powers, and vice versa. This both reduced the risk of war with Japan, and increased Britain's margin of naval superiority in the event of war. If war broke out in Europe, the RN could comfortably move ships from the Pacific to more important areas, like the Mediterranean. Finally, the RN recognised that the Russian loss at Tsushima did not resemble a situation it was likely to find itself in. The Russian Baltic Fleet defeated there had been caught at the end of a long voyage - the RN, with a more widespread network of bases, and unlimited access to the Suez Canal, would not be in a similar situation. The Russian fleet was poorly trained, and lacking in morale and maintenance. The RN was generally better-prepared for a battle. However, on a more strategic scale, it did lead to a realignment of the RN's priorities. With alliances secured with France and Japan, Russia's fleet neutralised, and the USA generally friendly, Germany was the key remaining threat. The RN began to pull units back from outlying stations, like the Pacific, to Europe.
As far as what the RN learned from the Russo-Japanese War, this is a somewhat complicated situation. The Admiralty had been working to transform the Navy in the years leading up to the war. The dreadnoughts were based on ideas that a number of naval architects, including Jackie Fisher and Carlo Cuniberti, had been developing in the late 1890s and early 1900s. Design work for the Dreadnought began well before Tsushima, and she was laid down slightly before the battle. With the RN developing a new paradigm in naval warfare, it felt it could learn little from the Russo-Japanese War, which it saw as outdated. However, it was also well-informed about the war, having several observers aboard Japanese warships thanks to the Anglo-Japanese Alliance. Gunnery was one area of particular concern, confirming the RN's decision to switch to an all-gun armament for Dreadnought and following battleships. The RN's most senior observer, Captain Pakenham, would describe Tsushima as follows:
The 10in guns of the Peresviet and Pobeida were of 45 calibres, and may also be of greater range, but the effect of every gun is so much less than that of the next larger size, that when 12in guns are firing, shots from 10in pass unnoticed, while, for all the respect they instil, 8in or 6in guns might just as well be pea shooters, and the 12pdr simply does not count. This must be understood to refer entirely to the moral [sic] effect.
There was considerable debate between two major factions within the navy; those arguing for all-big-gun ships like Dreadnought, and those arguing for 'a hail of fire', enabled by having more, smaller guns, as on the pre-dreadnoughts. Tsushima gave good arguments for both sides; fought at close range, it seemed like the volume of fire from the smaller guns was more effective. However, through closer examination of the damage done to Russian ships, as well as prisoner interrogations (which Pakenham's observations were likely based on), it became clear that the larger guns were much more effective. Earlier actions, especially the Yalu Sea action on the 10th August, showed that naval gunnery was capable of much longer ranges than had previously been envisaged, lending further weight to the all-big-gun side. Pakenham argued that the RN was capable of engaging at 20,000 yards or more, ranges previously unheard of. The RN also observed how Japanese gunnery caused Russian ships to sink. Pakenham noted that the long centreline bulkheads used on Russian ships made them unstable when flooded, causing a number of capsizes; however, this was ignored by the RN, which instead blamed it on the extreme tumblehome hulls used by the Russian ships. Attention was also paid to the need to reduce the spread of flooding, by reducing the number of pipes, ducts and doors piercing bulkheads. The risk of this was noted, but it was felt that actions taken in light of the loss of Victoria to an accidental collision would be sufficient. The dangers of fire were similarly noted, but again, it was felt that actions already taken were sufficient. However, one important lesson was not learned. The Japanese used shells filled with Shimose, a highly sensitive explosive much like the RN's Lyddite. Both fillings were likely to be set off by the impact of the shell on armour plate, greatly reducing the ability of the shell to penetrate armour. Japanese Shimose-filled shells did not penetrate Russian armour once during the war, while British Lyddite-filled shells struggled repeatedly during WWI.
The other major threats that the Russo-Japanese war revealed were the threats from mines and torpedoes. During the opening action of the war, the Japanese launched a surprise attack on the Russian Pacific Fleet, scoring three torpedo hits on Russian ships there, the battleshipsRetvizan and Tsesarevich and the cruiser Pallada. The Royal Navy felt that Tsesarevich had been particularly resistant to damage from the torpedo hit; they attributed this to her thick inner bulkhead. Tests of a similar scheme were quickly carried out, and this was soon incorporated into Dreadnought. However, this was likely a misreading of the situation, as the torpedo hit probably occurred aft of the bulkhead. Both sides took heavy casualties from mines, with the Japanese losing two battleships, four cruisers and five smaller ships, while the Russians lost a battleship, and another heavily damaged on two separate occasions. The RN had previously paid little attention to minesweeping, but the events of the war shone a harsh light on this neglect. The first RN experiments in minesweeping began in 1907, and an effective sweep was developed in 1908. The development of a minesweeping capability was somewhat neglected, as it was planned to requisition trawlers for use as minesweepers rather than developing purpose-built ones. This policy was to lead to significant problems at Gallipoli.