Millions of years in any one place, but probably not longer than a few hundred million for the whole galaxy - the starbursts will happen in different nebulae at different times. The radiation from starbursts (at a distance) won't affect us very much; we have not yet shielded ourselves from cosmic rays, and two fifths of cosmic rays that we experience are from our own sun. Still, we have flown out to our moon and back. Doubtless the radiation will take its toll, but I doubt that we on Earth will see that significant an increase in radiation from the Magellanic clouds, or for that matter if we were travelling around the local group of stars.