The process was remarkable in how quickly it happened.

In 1854, the Russian fleet destroyed the Turkish fleet in 1854 at the Battle of Sinope using explosive shells. This type of shell allowed the Russians to destroy the Turkish fleet in less than an hour. The explosive shells quickly set the wooden ships on fire.

Both France and Great Britain decided to build an ironclad vessel to counter this use of explosive shells. France lead the race by beginning the Gloire. Great Britain settled on a design based upon countering the Gloire. Both of these were basically modifications of the existing wooden ship designs. They had sails and had steam engines to supplement the sail. They were configured with large numbers of guns configured in a broadside, just like existing wooden gunships.

Great Britain's first design was the HMS Warrior. It was not completely armored. It had a bastion style armor where the lower gun decks, magazines, and engine room were armored. The armor did not extend to the bow or stern. The upper deck was also open. They had other design ideas, such as using turrets instead of a broadside configuration, but they decided on a more conservative design.

The idea was that it would operate in a line of battle.

Then, the American Civil War happened. Both sides in the Civil War immediately began building ironclads. The United States decided on three different designs. The New Ironsides, the Galena, and the Monitor. The United States also began to convert existing riverboats to ironclads for the combat along interior waterways. Of the three designs for ocean going ships, the New Ironsides was basically the same design concept as the HMS Warrior. The Galena was a smaller version, but was inferior in almost every metric. The Monitor was extremely radical design being a very small vessel tge rode very low in the water. It had no sails and was powered by steam only. It used a turret with 2 massive guns as it's gun battery.

The Confederacy settled on a strategy of also building ironclads, but had a far less well developed industrial base. Their first ironclad ocean going ship was a salvaged US Navy ship. They cut off the superstructure and built what amounted to an armored shed with the gun batteries on the above water portions of the ship. (The Confederacy used this design as their base design called casement for the entirety of the war). This ship was named the CSS Virginia.

At the Battle of Hampton Roads, the CSS Virginia destroyed the USS Cumberland and USS Congress in a short engagement. The following day, the Monitor arrived in Hampton Roads and engaged the Virginia in an hours long inconclusive battle.

(There were many other battles in the Civil War and they affirmed the basic outcome. A wooden ship was completely outclassed and the only possibility of a wooden ship defeating an ironclad would be swarm tactics and accept the loss of some of the attacking ships. The ironclad CSS Albemarle was able to engage 8 wooden ships and the battle was inconclusive, even though the Albemarle was hit by over 500 shells).

But, the Battle of Hampton Roads is the key here. It was front page news in Europe. While the battle was inconclusive, the battle between ironclads and wooden ships was not inconclusive. As the construction of the ships destroyed was similar to British ships of the same class and the guns were comparable, tge British government quickly reached the conclusion that their wooden gunships would not be effective against ironclads.

The British Admiralty rated the CSS Virginia as the better overall design, but the Monitor construction had several traits that were of greater concern. The Monitor was designed, built, launched, and engaged in combat in less than 4 months and for $275,000. The ship was small and was designed to use guns that would allow it to destroy much larger ships. The United States could crank out a fleet of ironclads that the British navy could not defeat with wooden ships. And, the level of shipbuilding was not out of reach of a large number of other countries, such as Peru, Brazil, and others.

So, the British navy considered its navy to be obsoleted against any industrialized opponent. They stopped contracts on wooden ships being constructed and switched to several ironclad alternatives. They funded several research programs to test technologies. To be fair, the British had already made the decision to move to all metal warships in 1861, but had planned a phased approach. That was scrapped. It became an Ironclad Now program and the major navies, such as France, Spain, and Russia were all building ironclads by 1862.

Wooden ships were of only marginal use after the appearance of ironclads. It went fast.

Reference: Clad in Iron, The American Civil War and the Challenge of British Naval Power by Howard Fuller