Just a brief correction to your question before I answer it:

Tea has a variety of health benefits, almost too many to list here, however I'll just note that black tea helps asthmatics tremendously, and tea has been drank for medicinal reasons since time immemorial. It suppresses appetite, contains fluoride, raises alertness and is a powerful source of antioxidants.

Tea came to the Dutch first in 1610. The Wuyi mountains of northern Fujian Province began producing black tea and sending it down river to the trading port of Canton, where the Dutch were furiously buying Chinese products.

Wealthy Dutch patrons fell in love with the drink immediately, and the wealthy important people from The Hague took to tea with a passion.

The addition of milk to tea was based on a rumor that the Manchu emperor at the time took his his tea heavily laced with milk. It is highly unlikely that any Han Chinese emperor took tea with milk, but it didn't stop Dutch physicians from recommending milky tea as a curative medicine.

Once the Dutch upper class started the tea habit, the French upper class did as well, thanks to vigorous Dutch trade in all things Chinese. The French referred to it as "tea a la Chinoise" (tea with milk).

By 1650, it became popular in Germany as well.

But it was not until 1658 that a popular coffeehouse in London called "Garraway's Coffee House" began selling Dutch-traded, Chinese tea.

And the English went mad for it. It was once again, touted as a healthful curative. It became fashionable for the wealthy to take tea, and was added to to the menus of houses that served hot chocolate and coffee to the privileged class, professional men and "Literati".

When Charles II wed Princess Catherine of Braganza from Portugal, tea became the pastime of English ladies, and the rise of "Tea time".

With the rise of the sugar trade from the West Indies, the English continued the Dutch tradition of drinking tea with milk and sugar, and it was practically prescribed as a twice-daily ritual for health.

So to answer your question, it became a popular health tonic for the wealthy in Europe, and trickled down to the masses as tea trade increased.

My understanding is it partially started when the church of England split from the Catholic church. Coffee was strongly associated with the Catholic church at time, and to distance themselves, people in England began to stop drinking it, and instead started drinking tea. There is actually a theory that tea helped out the industrial revolution, because it has minor antibiotic properties. Right when people started really bunching up in cities, is when tea got popular. It also may have contributed to British naval superiority, due to it's vitamin C. This helped fight off scurvy, and major problem at the time.

I would check out this book http://www.amazon.com/A-History-World-6-Glasses/dp/0802715524 If you are interested in more information. It goes too far to the side of "this caused this" but as long as you keep in mind things are rarely as cut and dry as he implies, it has some great information.