From my perspective, that's like asking if you're pleased 2 muggers were fighting over who got to have their way with you first and one chased the other off.
That said, I think your assertion that world leaders have "breathed a huge sigh of relief" because they were "terrified of the Bush mindset that a Romney victory would have wrought" is wrong.
Obama has continued and expanded virtually every one of Bush's policies, especially his foreign policy. Gitmo is still open. We are still fighting endless interventionist wars and meddling in the affairs of other sovereign nations. We have a military presence in more nations around the world than at any other time in our history.
Domestic policy is much the same. The police state continues to grow. The Patriot Act is still in effect. Obama signed the NDAA into law, which allows for the indefinite detention of American citizens without due process. Obama has revved up the War on Drugs - championed and supported vehemently by many on the right - and our prisons are full of people whose only "crime" is possessing a plant (but there is a glimmer of hope in that area with Colorado and Washington legalizing pot for recreational use). We continue to reward failure by taking money from the middle class and bailing out big banks and corporations instead of allowing them to fail.
And the stark reality is, Romney's policies would have been essentially the same. No matter the outcome last night, George W. Bush was guaranteed to win his 4th term.