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Most of the wooden ships remaining in the RN suffered one of two fates. They were either broken up, or repurposed for second-line roles as training ships and the like. The former tended to be done to smaller ships like frigates, while the majority of the ships in the latter role were the larger ships of the line. Ships that were broken up were often just set on fire or scuttled. However, it became increasingly common for the ships timbers to be recovered, as they were made from fine, hard-wearing woods. Castle's yard on the Thames at Millbank was a major pioneer in this - Turner's famous The Fighting Temeraire depicts the eponymous ship being towed to their yard.
The RN had a major desire for old ships to use as fixed installations in harbour. These ships had their masts removed, turning them into hulks. They would then be used for a variety of purposes. The 104-gun Dreadnought was converted to a hospital ship in 1831, a role she'd later be replaced in by the 120-gun Caledonia. Many hulks were used as accommodation ships or receiving ships, housing sailors until they could be assigned to a ship in the fleet. Other ships became training establishments - Queen Charlotte became HMS Excellent, the RN's gunnery school, while the Donegal and Marlborough formed the RN's torpedo school, HMS Vernon. Ships in this role could survive for years - HMS Victory served as a signal school during the late 1800s and early 1900s, while Implacable, captured after Trafalgar, would be scuttled in 1949 following a varied career as a training ship, accommodation ship and coal hulk.
That said, wooden ships would linger in second line combat roles - the sloops of the Osprey and Doterel classes, built in the late 1870s, were constructed with a wooden hull over an iron frame. They were designed to patrol British colonial possessions in the Pacific.