In a nutshell… (read this for a quick overview)
Unlike India, Sri Lanka’s history with tea is indissolubly linked to that of coffee.
In 1869, the coffee industry was still thriving in Ceylon, but shortly afterwards, coffee plantations were devastated by the fungal disease Hemileia vastatrix, also known as coffee leaf rust (CLR), or nicknamed by the planters "Devastating Emily”. 
The production of coffee dipped rapidly and by 1900, coffee was only being cultivated on 11,392 acres (46 km2). Its successor was the humble tea plant that was not affected by CLR .
A bit more long winded…
The British gained total control of Ceylon (Sri Lanka) from the Dutch in 1802 through diplomatic means outlined in the Treaty of Amiens before the start of the Napoleonic Wars.  Like India, Sri Lanka’s relationship to tea evolved under British rule.
Coffee was a common crop of the native Sinhalese, having been introduced to the island years before by Arab traders. The Dutch had tried their hand at cultivating it with little success due to living at lower elevations of the island where the coffee bush suffered with the humidity and heat.
In 1825 the British started developing a scientific and commercial approach to the cultivation of coffee. Using a bit of common sense they moved the coffee bush to the highlands where the temperature was cooler, thus allowing the plant to flourish. This was a difficult move from the low lands to higher inland areas as workers would have to slash jungles, brave wild leopards and withstand tropical diseases in the noble quest to plant more coffee bushes. Many of the government incentivised workers were Scots who purchased land, planted coffee trees and expanded their holdings as time went on.
As the plantations increased and more labour was needed the workers were brought in from the Tamil population of southern India to work the fields. The excitement and hype about coffee in England fueled speculative investments and then a market crash, historically known as ‘coffee mania’, spanning the years 1830 to 1845. 
During the rush for coffee, there were some that experimented with tea cultivation. Beginning with the Dutch in late 1839 seeds from Assam tea bushes grown by Dr. Nathaniel Wallich of the Royal Botanic Gardens in Calcutta were sent to the Botanic Gardens in Ceylon. Thereafter, 250 Assam tea bushes followed. Many coffee planters used these seeds and plants to begin experimenting with tea through the 1840s and 1850s. It was these coffee planters who eventually became the leaders in the tea industry in the 1880s after the coffee blight totally wiped out the coffee plantations.
The first mention of a strange orange-red blotch appearing on the underside of coffee leaves was in 1860. By 1869 the coffee blight was spreading rapidly from one coffee estate to the next. The peak of the coffee prosperity on the island in 1877 was short lived, as the coffee blight, also known as coffee rust, left the thriving coffee plantations in ruins. Many European planters and Tamils alike fled the island, rather than face the devastation. But there were those who stayed.
According to William Ukers, “There was left only a small band of planters who made a pathetic picture standing together amid the ruins of their fortunes, but stubbornly refusing to accept defeat. To these men who faced seemingly insurmountable obstacles at the blackest period of Ceylon’s history, the tea trade of the world is indebted for the raising of a magnificent industry from the ashes of the coffee estates – an industry conceived in penury and nurtured in economy, yet producing today the finest quality teas that reach the world’s markets.” 
So dire was the coffee plantation situation, that dead coffee trees were stripped of limbs and the logs were exported to England to serve as legs for tea tables In order to gain any monetary compensation back.
To this day one the oldest tea fields under continuous cultivation is Loolecondera, situated in the growing province of Kandy. At the time a Scot named James Taylor (1835-1892) who had spent most of his adult life working at Loolecondera coffee plantation was to set to lay the foundation for Ceylons Tea industry. The owner of the plantation instructed the young manager to plant Assam tea seeds on a trial basis.
The trial went well and based upon a report by an agent from the coffee planter’s association, Mr. William Martin Leake, (one of the plantation owners) ordered more seeds from Assam to be planted on the Coffee estate. James Taylor was asked to clear 20 acres fo make way for the new seeds.
During this time James Taylor installed a rudimentary but practical tea manufacturing facility at Loolecondera and produced the first Ceylon tea, which was sold in Kandy in 1871. In his words “"I have a machine of my own invention for rolling tea which I think will be successful". In 1872, 23 pounds (10kg) of black tea were shipped London for sale. After visiting the Darjeeling estates in 1874 and with new knowledge, by 1890 the estate was producing 22,900 Tonnes. James Taylor met with Sir Thomas Lipton in Ceylon during the 1890s to discuss the business of tea in Ceylon and London. 
Around this time tea production spread throughout the beautiful island nation and in 1875 one thousand acres of once profitable coffee terrain was converted and planted in tea. By 1895 over 300,000 acres had been planted in tea. That amount increased to 467,000 acres by 1930.
The rapid growth of the Ceylon tea industry allowed the large already established tea companies to take over therefore consolations were inevitable and James Taylor was laid off by the Loolecondera estate management.
James Taylor unfortunately died in 1892, just one year after his dismissal from Loolecondera estate, from severe gastroenteritis and dysentery. He was buried in the Mahaiyawa Cemetery in Kandy. His headstone reads, "In pious memory of James Taylor of Loolecondera Estate Ceylon, the pioneer of the cinchona and tea enterprise in this island, who died 2 May 1892, aged 57 years". He is considered the father of tea in Sri Lanka today and a museum was built to commemorate his life’s work in 1992. (Ceylon became Sri Lanka in 1972.)
In 1893 one year after his death, one million packets of Ceylon tea of the first shipment to London were sold at the Chicago World's Fair. All of the tea produced in Ceylon was black tea until it was pointed out by the Ceylon tea commissioner to the United States in 1895 that Americans drank green tea. The manufacture of green tea commenced in 1899, encouraged by a stimulus package that provided a bonus for every pound exported.
The tea industry in Ceylon faced and survived several market challenges in the early 1900s. When they themselves caused a market slump by overproducing low quality teas and having insufficient storage.
Taking notice of the global tea situation they recognised that high-quality teas would be much less likely to suffer from overproduction and decided to focus on producing high-quality teas. They still faced challenges from the global market through the years, but Sri Lanka has maintained its distinction in the world market by producing high-quality loose-leaf black teas.
A nation synonymous with tea, build upon the history of coffee, Sri Lankan Coffee is making a comeback. Thanks to Lawrence Goldberg, an expatriate American who has recently partnered with an Australian. The revival is underway with most of the coffee produced going to Sri Lanka high end hotels.
Goldberg runs a modest cafe in Colombo, where you can order a great caffe latte (flat whites haven't made it to the Sri Lanka as yet) while Nuwara Eliya is the home of his company, Hansa Ceylon Coffee. It bills itself as Sri Lanka's first gourmet coffee for 150 years specialising in quality Arabica and robusta beans.
Hansa Ceylon Coffee was originally founded by a Dutchman in the mid-1990s with the enterprise partly funded by the Netherlands government as a developing world aid project but it's since evolved into a proper going concern.