How does the no-cloning theorem allow to replicate or copy any kind of information? I am a recent undergraduate from India and currently studying quantum mechanics.
I came across a result that quantum mechanics doesn't allow the cloning of even the smallest atoms because that would break the uncertainty principle.
I have a doubt that if we believe that nature is quantum mechanical and follows the postulates of quantum mechanics, how I am able to replicate or copy any kind of information, say backing up my hard drive? Correct me if I sound naive because I am just learning.
 A: The no-cloning theorem does not forbid cloning classical information, even if it is stored in a single quantum system.  For instance, you can perfectly copy classical information which you encoded in a single photon by using horizontal and vertical polarization.
Thus, cloning of classical information is perfectly with also within quantum mechanics.  This includes your hard drive, which you use to store classical information.
What the no-cloning theorem precludes is only copying arbitrary quantum information (that is, information encoded in states which cannot be classically distinguished).
A: Backing up your hard drive does not involve making an exact copy of all the fundamental particles in your hard drive and their quantum states. It only involves making an approximate (low fidelity) copy that is good enough so that the same information can be retrieved from the copy and from the original.
The no-cloning theorem does not say that information cannot be copied or duplicated. It very specifically says that if system $A$ is in an arbitrary quantum state then its quantum state cannot be exactly copied (without disturbing system $A$) into the quantum state of some other system $B$.
