If the conscious awareness and free will of human beings can aim arrows and bullets to produce desired results, then our creator could also aim the radiation particles to produce the desired mutations.
So you have an alternate theory of everything in which things only appear to be natural but under the surface God is secretly running the show.
So this God creates a cosmos such that life evolves and speciation occurs but it is not qute going according to divine plan so he has to constantly interfere with it by precision aiming charges particles from deep space such that they alter the course of life.
Which kind of begs numerous questions :
Why didn't he get it right in the first place ?
Why use seemingly natural processes at all if he is going to have to use his god magic to steer things in the end anyway ?
Why be secretive about the whole business, apparently only expressing his powers using 'quantum events' or charged particles from deep space so that noone can discern that these events are not entirely natural ?
If he uses this mechanism to bring about beneficial mutations why has he not also ridded our genomes of all the bad stuff, the thousands of heritable conditions still present in the human genome ?
Apart from whch, your scenario is insufficient - such mutations would have occurred naturally anyway, it is the selecting nature of the environment which determines whether mutations are conserved; so you would have to expand your rationale to say that God intervened in the environmental niches, altering the balance points and population dynamics and predator/prey relationships such that his divinely inspired mutations propogated through the population.
All of which leaves us wondering why this God is 1/ so secretive, and 2/ why didn't he just create things as he wanted them in the first place.