During the Napleonic wars the British used the Congreve Rocket, a design inspired by Mysore rockets the EIC had faced in India.
But what happened to these rockets after the Napoleonic wars?
Why weren't designs for rocket artillery continually developed in the same way conventional artillery was?
Congreve Rockets had two big advantages over Napoleonic-era conventional artillery — rate of fire and ease of transport. But they were never terribly accurate and generally had rather pitiful range (less than 1 mile). Although slower to fire, canons and howitzers could reach targets further away and with much greater precision.
As for their usage post-1815, the British Army continued using Congreves until the latter part of the 19th century. The weapon type was deployed in a number of campaigns, including the Crimean War, the suppression of the Indian Mutiny and the Anglo-Zulu War. The advent of the breech-loading Armstrong gun in the middle of the century (which vastly improved the rate of fire for conventional guns) pushed the Rocket towards obsolescence, although they were completely phased out from British armouries in the 1880s.
Further reading/info
Harold E. Raugh, The Victorians at War, 1815-1914: An Encyclopedia of British Military History
Simon Werrett, "William Congreve's rational rockets"
A. Bowdoin van Riper, Rockets and Missiles: The Life Story of a Technology
As u/booksworm says, artillery really went through a revolution in the late 19th Century with breech-loading, hydraulic recoil weapons often firing a combined shot-and-cartridge round. This set artillery arms in a path of developing methods and tactics to maximise their effectiveness. During the First World War, techniques centred on achieving greater accuracy and 'choreography' of bombardments- rockets were of little use to this end.
In the Second World War rockets underwent a renaissance- they were cheap and could fire a large warhead, while mechanisation improved their portabiity. For the USSR and Germany, rockets were simple to manufacture, required less training to use, and could 'scale' up and down. Rockets could be set up, fire and move (because the projector was light- no heavy steel barrel) which was helpful if air attack was likely, or counter-battery fire.
Rockets proved the answer to new problems such as how to carry an explosive weapon on a plane, or project a shaped charge to defeat a tank, from a weapon the infantry could carry.
The West with its vast industrial potential, and air superiority allowing emplaced artillery to survive, still tended to use conventional artillery- with staggering effectiveness- but rockets have stayed around in a secondary role ever since.