I think "underway" also has a sense of that there is an end and definite purpose in sight, while "ongoing' implies something regular and normal that doesn't have a clear beginning and end.

"This court case is underway" means it has just started "This court case is ongoing" implies that it's been going one for a while with no end in sight.

Example: Finally this work is underway! It took so long to get it started. This road construction has been ongoing through the whole summer. But as I said, the meaning is somewhat overlapping.

Underway, they have started and they are progressing Ongoing, they are continuing. There is an overlap in meaning, but underway usually stresses the fact that it started and keeps going, whereas ongoing usually stresses that it is a continual work that has been going on for a while.

You are welcome! Happy studies. : )