I guess it would depend on whether Smaug would have been acting on his own or at Sauron's behest.
Smaug himself might be sufficiently interested in Bilbo for goblet-thieving and being a sneaky annoyance to go after him, but he wouldn't have known where Bilbo came from (except presumably somewhere west of the Misty Mountains, from what Bilbo told him in his riddles), and I don't see
him inquiring for directions to the Shire like the Black Riders did, so the Shire might have been safe, for a while at least.
If Sauron would have controlled Smaug somehow, I doubt he would have sent him for the Ring instead of the Nazgûl, knowing how possessive dragons tend to get about treasures. But he certainly could have used Smaug to attack his other enemies - Thranduil probably first, then Lórien, possibly Rivendell, maybe even Gondor and Rohan. Like Tolkien says in LotR Appendix A III (Gandalf speaking):
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Think of what might have been. Dragon-fire and savage swords in Eriador, night in Rivendell. There might be no Queen in Gondor. We might now hope to return from the victory here only to ruin and ash.
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