it seems that the British Admiralty hardly gave a second shit about Tirpitz.

On the contrary, the Admiralty was quite concerned about the potential of the Tirpitz. When the battleship deployed from Germany to Trondheim, Norway in January 1942, Churchill directed both First Sea Lord Dudley Pound and to Bomber Command that the sinking or crippling of the Tirpitz was essential for the conduct of the war in the Atlantic.

The problem for the British was that both the loss of the Bismarck and the Kriegsmarine's crippling fuel shortage meant that the Tirpitz largely spent the war in the Norwegian fjords. Admiral John Tovey, the commander in chief of the British Home Fleet outlined the problem in a memorandum in March 1942:

They are therefore most unlikely to risk the Tirpitz except for an objective of outstanding importance. German heavy ships have never sought action with ours, however weak. I consider it possible that Scheer and Prinz Eugen were sent to Trondheim with the idea of raiding Russian convoys but most improbable that Tirpitz would take part except possibly in support and still more so she would accept action if heavy ships were covering the convoy.

Both Tovey and Pound decided to start adding a large surface force screen to Soviet-outward and inward bound convoys to lure out Tirpitz and whatever other German heavy ships that tried to disrupt the convoy. When the Kriegsmarine received permission from Hitler to sortie Tirpitz to attack convoys PQ 12 and QP 8 in March 1942, Tovey turned his heavy cover force consisting of two battleships, a battlecruiser and an aircraft carrier towards the German ships. Bad weather prevented the both sides from making contact, but HMS Victorious was able to launch an ineffective airstrike against the German ships. In the meantime, Bomber Command began to periodically raid Trondheim but the range and stiff German defenses also rendered these attacks ineffectual. The threat of a Tirpitz deployment in July 1942 caused the withdrawal of the escort forces for convoy PQ 17 to supplement the covering force and a scattering of the convoy, leading to a disastrous convoy battle where German air and u-boat forces were able to pick off 24 of the convoy's isolated ships.

The failure of the RN and RAF's conventional forces to deal a crippling blow to the German battleship meant that they turned to special craft to do the job. Although an attack using Chariot manned torpedoes failed in October 1942, the larger X-craft midget submarine attack on September 1943 severely damaged the ship. The RAF began to train crew for using its Tallboy bombs, which were designed to destroy hardened targets, to attack the German battleship. In the meantime, the RN launched six conventional carrier strikes against the Tirpitz in 1944 which further damaged the ship. The RAF's Tallboys finally did her in over the course of three air raids between September and November 1944.

So the British did commit serious resources to sinking the Tirpitz. Unlike her more famous sister, the Tirpitz largely stayed in port as a fleet-in-being, which hampered the British response to the ship. The few times she did sortie, or appeared to, she did provoke a counter-response for the RN, but by staying in heavily-defended Norwegian ports, the British had to come up with a measured and judicious plan to neutralize the Tirpitz threat.

There's a few misunderstandings in this question - Force H wasn't in or anywhere near Crete at the time. They had been providing cover to a Malta convoy before Bismarck's breakout, but they were always in the Western Mediterranean. After all, the point of the force was to harass Italian forces in the Western Mediterranean, and to prevent their breakout through the Straits of Gibraltar. In addition, the RN and RAF put a large amount of effort into counteracting Tirpitz. Nearly every convoy to the Soviet Union had at least one battleship (usually two) acting as a distant escort. Home Fleet had to maintain a strong battleship strength to prevent any action by her, diverting them from other theatres where they could be more useful. Some would also be supported by aircraft carriers. The RAF made multiple bombing raids against the ship, culminating in the attacks which sank her. Carriers of Home Fleet launched three operations involving raids against the Tirpitz at anchor in Kaafjord, Operations Tungsten, Mascot and Goodwood. Then you have the raids by RN miniature submarines and human torpedoes, and the commando raid on St Nazaire to deny her a base on the Atlantic coast of France.

However, the primary reason for why Tirpitz survived for 3 years compared to Bismarck's week was that Bismarck posed a much bigger threat to the UK than Tirpitz ever did. Bismarck was cruising about the North Atlantic, threatening to wipe out any convoy she ran into. If she made it back to France, she would still pose a substantial threat to the Atlantic convoys, especially as she would have joined up with the battlecruisers Scharnhorst and Gneisenau. In comparison, Tirpitz was threatening the convoy routes to Murmansk and Archangelsk. These were important to the UK's war aims, but nowhere near as vital as the Atlantic convoy routes. Had the Arctic convoy routes been shut down, lend-lease could still have flowed to the Soviet Union through Persia or the Pacific. There was no way to replace the Atlantic convoys, which didn't just carry lend-lease equipment and American troops. Britain imported much of its food at the time, and this had to come across the Atlantic as well. Tirpitz was a threat, but not an existential threat as Bismarck was.

You also have the impact of the Battle of the Denmark Strait providing additional impetus for a rapid sinking of Bismarck. The Hood, one of the largest and most prestigious ships in the RN, had been sunk in minutes. British admirals wanted revenge for her sinking. This led to the firepower of King George V and Nelson being concentrated on Bismarck's superstructure during her final battle - the RN had caught her, and wanted to do as much damage to her as possible before she sank as a measure of revenge.