[Child of the 7th Age]

As you've said, the Hobbit and the LOTR are definitely different in tone. However, if you read the beginning chapters of the Rings, especially A Long-Expected Party, you'll see that the change is actually gradual. When Tolkien first began writing the "sequel" requested by his publisher, he really didn't know where the story would lead him. He didn't recognize at first just how high and serious it would become, and how it would tie into many of the important themes of the earlier ages.

So the earliest chapter starts out sounding almost like a continuation of the Hobbit. It's got a lot of humor, for example poking fun at the foibles of some of Bilbo's relatives and neighbors by the particular kind of presents he left for them which highlighted their shortcomings. Also, it has a very colloquial style because that's just the way most Hobbits speak (though the speech of Merry, Pippin Frodo, and even to some degree Sam changes later in the book as they interact with more courtly cultures.)

In the Hobbit, the Shire often sounds more like it belongs in Victorian or Edwardian England rather than the ancient, heroic cultures like the men of Gondor or Rohan. One example of this from the Hobbit would be when Bilbo is all concerned because he lost his handkerchief, an item that figures several times through the story. There is some of this Victorian/Edwardian flavor still left in the first chapter of LOTR. When they describe the dragon fireworks, for example, Tolkien says it "passed like an express train" which is obviously not something you'd find in Middle-earth.

Of course, some of this simply reflected the simplicity and simple joys of Hobbit culture. But, by the time you come back to Hobbiton at the end, with the Scouring of the Shire, there's a much different tone.

So even though there's a sharp contrast between the two books, the actual transition in tone and language comes about more slowly.