Conceptual confusion about NMR
The nuclei absorb the rf radiation and the spin population in the upper level increases at the expense of that in the lower level. There is now a chance that the upper levels will radiate away their extra energy and so return to equilibrium. However, in an nmr experiment the excitation pulses used in most if not all nmr machines are very short, just a few microseconds (and so spectrally wide) but the natural radiative lifetime of the excited levels is some seconds. Thus it is possible to excite the nuclear spins and build up a relatively large population before any significant decay occurs. 

(This is one of the reasons why nmr works so well considering the tiny population difference that is produced because the energy levels are so close in energy and so all levels are significantly thermally populated.  The long lifetime of upper levels means that it is possible to coherently build up a significant macroscopic magnetisation in the sample and this what is detected.)