For previous entries in this series, follow the tag at the end of the post.
Say the word "knight," and it conjures an image of a man in plate armor, ready to joust for the honor of his lady fair. But there were still knights in the 18th and 19th century (and to this day). Just minus the armor.
A knighthood is an honor given to untitled men (but in the 18th and 19th centuries generally of at least genteel birth) as a reward for service to the Crown. I haven't done a detailed study, but my impression is that most reasonably successful generals and admirals of the Napoleonic era were at least made knights. Wellington's first major honor was being knighted in 1804 in recognition of his early successes as a major-general in India. So from that point until he was granted a peerage in 1809, he was addressed as Sir Arthur Wellesley. As with the younger son's courtesy titles we've discussed in the past two weeks, the "Sir" goes with the first name. He's Sir Arthur, not Sir Wellesley.
Wives of knights, however, don't follow the same pattern. In 1806, Wellesley married Catherine Pakenham, whom he'd courted as a young man before leaving for India. In the intervening time they'd grown into different people, extremely mismatched different people who had a thoroughly unhappy marriage. But that's neither here nor there. We're just here to learn what to call them. And in this case the right answer is Lady Wellesley. Not Lady Catherine, not Lady Arthur. Lady Wellesley.
Incidentally, at this point there were two Lady Wellesleys. (Ladies Wellesley?) The other, Hyacinthe, was married to the oldest Wellesley brother, Richard, who'd been granted the title of Marquess Wellesley for his service as Governor-General of India, superseding his former title of Earl of Mornington (inherited from his father). This situation wasn't as confusing as you might think, for two reasons. The first is Hyacinthe's background: she was a French actress who was Richard's mistress for years and years before he actually married her. As such, despite her status as marchioness she wasn't accepted in good society or even within the extended family the same way that the well-bred and well-behaved Catherine was. The second is common sense--just like you might have two friends named Bob Smith, and avoid any confusion by talking about Work Bob and Bob from High School, English society of 200 years ago was perfectly capable of saying "Lady Wellesley, the marchioness," or "Lady Wellesley, Kitty Pakenham that was," or whatever to make their meaning clear.
Note that knighthoods are not inherited honors. If Wellesley had been killed while he was still a knight, his firstborn son would NOT have become Sir Arthur in his turn.
A side note about baronets: While I don't have any Wimsey or Wellesley examples of them, you'll see them a good bit in 18th and 19th century fiction. In Jane Austen's work alone, Sir Thomas Bertram in Mansfield Park and Sir Walter Elliott in Persuasion are both baronets. Baronets follow the exact same form of address as knights--the key difference between them is that the eldest son of a baronet DOES inherit the honor.