There are two issues here that you need to understand. First, you need to know the difference between left and right wing attitudes towards change. Next, you need to know the difference between a radical and a reactionary.
The question you have posed implies that this person is saying "no" to change that has not happened, but which inevitably will happen. Such a person cannot be a reactionary because a reactionary is someone who wants to go back to a state of affairs that had existed before. The person you are asking about does not want to change. These are different things. So the person has to be a radical when a radical is defined as someone who is extreme in their attitudes.
This radical is probably a right-wing radical. This is because conservatism (the right of the political spectrum) is an ideology that does not readily accept change. Conservatives believe that the way things are in a society has been arrived at by years of experimentation and should probably be left alone. Therefore, they dislike change. A person who dislikes change as much as the person in your question would, therefore, be a right-wing radical.