How fast does a supernova blow up? I mean, we have atomic bombs here on earth and their detonation takes seconds to achieve full power, but this is a star blowing up, does it happen over minutes, hours, days or longer?

It all depends what you mean by "full power". Using your atomic bombs analogy, you should know that they release their full yield in a matter of nanoseconds, and that the resulting fireball is what takes a few seconds to expand fully. The actual explosion part, or nuclear reaction, is incredibly short.

Now, for stars, it's essentially the same thing. There are many different mechanisms for the appearance of supernovas, but they mostly follow the same pattern: inward pressure (caused by gravity) of the star rises faster than its outward radiation pressure, causing the quick fusion of hydrogen into helium, which continues for a few seconds or minutes, until the radiation pressure from released energy becomes greater than the inward pressure and blows out the star's outer layers. Like in atomic bombs, the "yield" of the supernova is released rather quickly (generally in the order of seconds or minutes), but the "explosion" looks like it lasts for a few years because in space, with nothing to stop it, the gas cloud ejected by the supernova explosion expands pretty much forever.

In the case of core-collapse supernovas, electrons from iron and nickel at the core of a star actually combine with their corresponding protons, which happens in a fraction of a second. The core collapses so fast that its outer layers reach a decent fraction of the speed of light as they fall towards their center, before rebounding outward and sending a powerful shock wave which breaks the star apart. Now, although the collapse of the core happens almost instantly and causes the core to contract at almost a quart of the speed of light, stars which undergo this kind of explosion are so large that the core collapses for a few seconds before rebounding and causing the star to explode. Once again, the outer layers take forever to expand, giving the impression of a long-lasting reaction.