This paper sheds light on the time structure of betatron radiation, emitted by electrons that undergo betatron oscillations as they accelerate under the action of plasma wakefields. It is a common practice to assume that the betatron pulses are as short as the electron bunch length, however we show that this is not a general rule. Indeed, the betatron pulse length is affected by the betatron motion, which stretches and modulates the radiation pulses already at the source level. Propagation in a vacuum, therefore, can greatly lengthen the betatron pulses by orders of magnitude. In the wake of the above, the coherent emission of betatron radiation is studied. Coherent betatron radiation has been found to propagate in an underdense region created by ponderomotive forces, thus not suppressed by the overdense plasma absorption. This could be observed experimentally, revealing information on the acceleration process and on key beam parameters.