When did the English and Americans realize that vegetables were healthy?

I conjecture that this was an empirical result deduced from experience with long distance sea voyages. Scurvy was a real disaster in many long expeditions. Seaman began the dietary experiments to fight scurvy in 18-th century (or maybe earlier, I just do not know the earlier records). At first they tried sauerkraut. The lemon juice was introduced to the sailor ration in 19-th century.

In the past there was a great diversity of opinions about what constituted a good diet and you can certainly find examples of people who deprecated, for example, vegetables. However, this was not typical. Overall, the most common recommendations were to eat a variety of food, and to eat less food. Many physicians going forward from Galen recommended the simple expedient of eating less, and eating a wide variety.

Just like today, there were many who recommended a vegetarian or vegetable diet, and cited the ancient Greeks or other supposed vegetarians, such as the "Hindoos" as their model, but once again these were in the minority. The average doctor typically prescribed the same thing doctors prescribe today: eat a variety of foods and reduce the amount you eat.

I am not sure if it is the first recognition of healthiness but about the end of age of discoveries, sailors started to realize if they bring sour cabbage with them, they can prevent various diseases. Or better to say some diseases are just don't show up on their ship for the long trip.

In those times they knew nothing about Vitamin C, and most of diseases were caused by serious lack of this vitamin.

I can provide this source about sour cabbage

Anson’s voyage brought scurvy to the public’s attention and after studying historical accounts of the disease, in 1753, Scottish naval surgeon James Lind noticed scurvy was invariably linked to those whose diet had been severely limited. He began testing various foods and noted that citrus fruits provided the quickest and most effective cure for the disease.

However, this brought about another problem. How do you keep fruit fresh on a sailing ship that could be at sea for months at a time? You don’t.

With no real cure available, the British crown outfitted four captains during the 1760s with various potential cures in an attempt to find a reliable method to prevent scurvy through trial and error.

Captain James Cook, one of these four captains, was given several different experimental foods to try aboard his ship the HM Bark Endeavor when he left England for the South Pacific in 1768. Among them, as noted in the victualing minutes — the log of provisions put aboard — was 7,860 pounds of sauerkraut.

So James Cook was one of the first captains who realized (without knowing the precise effect) of having consumable vegetables on his ship the health issues were way less serious on the long trips.

Looking into the other answers, it seems that weather and diet were closely linked in the minds of 19th century writers. Thomas Graham (see Semaphore's answer) may have been important in disseminating the belief that vegetables were most beneficial in hot climates, as this begins showing up elsewhere by the end of the century.

For example, in an 1864 trial in Illinois, there was debate over whether people who ate mostly vegetables would heal more slowly than people who ate mostly meat. The answer was "It depends on climate":

In 1888, "embarrassed housekeepers" struggled with setting a good table in hot weather. Mrs. S. T. Rorer published a book called "Hot Weather Dishes," which offered such sensible advice as:

Have some courses at every meal without meat, and learn to eat vegetables with bread. Most persons eat too much, and don't know that a good meal can be made of vegetables . . . To eat meat three times a day in Summer is barbarous.

This implies that by the late 19th century many housekeepers still served meat three times a day. Not knowing how to serve trendy vegetables, housekeepers were "embarrassed" in summer.

So by the late 19th century, conventional wisdom was beginning to catch up to the scientific advice of earlier decades. However, it seems that by the 20th century, professionals recognized the wisdom of eating vegetables year-round. For example, an English physician writing on health at boarding schools in 1905 notes that "as a rule, boys will not eat [vegetables] in the autumn and winter," and to this he attributes "autumnal eczema" and "constipation, which always occurs . . . on the advent of cold weather."

How long would it take for the idea that one should eat vegetables year-round to sink into the public consciousness? The motherly injunction to "eat your vegetables" only became commonplace around 1940, and really took off in the 1970s.