Hobbits are capable of moving with extreme stealth when they so choose, sufficiently quietly for this job. Here is a description from the book:
But at any rate hobbits can move quietly in woods, absolutely quietly. They take a pride in it, and Bilbo had sniffed more than once at what he called "all this dwarvish racket," as they went along, though I don't sup-pose you or I would notice anything at all on a windy night, not if the whole cavalcade had passed two feet off. As for Bilbo walking primly towards the red light, I don't suppose even a weasel would have stirred a whisker at it. - The Hobbit: Chapter 2 - Roast Mutton
In the movie, Smaug is indeed (absurdly) awakened by the tapping of a bird with a nut (or was it a snail?) In the book, it makes more sense. Smaug is awakened by the draft from the secret door, and by the hammering and general racket of the dwarves that had occurred in trying to open it.
[Smaug] had passed from an uneasy dream (in which a warrior, altogether insignificant in size but provided with a bitter sword and great courage, figured most unpleasantly) to a doze, and from a doze to wide waking. There was a breath of strange air in his cave. Could there be a draught from that little hole? He had never felt quite happy about it, though was so small, and now he glared at it in suspicion an wondered why he had never blocked it up. Of late he had half fancied he had caught the dim echoes of a knocking sound from far above that came down through it to his lair. - The Hobbit: Chapter 12 - Inside Information
The movie uses the bird's tapping as an illustration of Smaug's hyperawareness, but if we have to shoehorn it into a reasonable in-world explanation, then you could try something like the tunnel being a resonating chamber that amplified the bird's tapping.