Do you mean 'to sum up' and 'in conclusion' are used in situations such as a piece of writing or a speech, etc., and they are interchangeable and 'in a word' and 'in a nutshell' are used in situations such as a short and clear answer, explanation or statement in conversation, etc.?
Do you mean 'to sum up' and 'in conclusion' are used in situations such as a piece of writing or a speech, etc., and they are interchangeable To my mind, yes, they are. Although perhaps "to sum up" is a little less formal.
and 'in a word' and 'in a nutshell' are used in situations such as a short and clear answer, explanation or statement
Imagine you spent three hours in a movie theater. Imagine that the movie was terrible - just terrible. Would you feel it was a good use of three hours of your life? But unfortunately, you can't get them back.