Depends a bit on what you mean by that word "radiation". Radiation from radioactive elements is mostly electromagnetic (EM) energy ("light" even if it might be outside visible range) so redshift or blueshift will happen (doppler effect as it applies to light/EM radiation, which moves as a waveform of fixed frequency).

If it is a particle emission (alpha or beta decay, say), then there is no waveform involved so there will be no change in perceived wavelength at impact with the particle (no wave at all). The impact will be less (or more if a head-on collision) forceful, is all.

There is always also some EM emission whether or not there is particle emission, and the EM emission will do the wavelength shift if the receiver is moving fast enough to cover a significant proportion of a wavelength during the time interval of the individual wave. If the receiver moves a third of a wavelength distance in the time it takes for a second wave to arrive, then the perceived wavelength will be reduced or increased by as much as 1/3 (maximum change if wave and receiver are moving in the same or opposite direction but less if the two are moving obliquely).

Redshift with light from stars and galaxies is how we know that the universe is expanding; the further away the source, the more it redshifts, and further away means more time since the light began the trip. How we know how far away the source is, well, that is a different problem.