We have difficuties with new situations with at least one of our kids, too. My son is 4.5 years and highly sensitive. He will take a long time (relative to most other kids) to adjust to new people and situations. My experience is that this is not usually fear or shyness, but just a desire to observe closely and move carefully. The less we push, the faster he moves foreward. It is good to leave him somewhat alone with new situations and not crowd him.
As for your particular issue, i think it might work if her previous gym teacher, whom she seems to like, would "pass her on" and be the agent in the transition to the new teacher, leaving you out of the game a bit. That way the transition may be a bit more natural to her, and there may be less of a loss in what she cherishes about her "old class". It would show continuity and be honest about the change at the same time.
I think it is interesting to see what she cares about. It's not so much the gymnastics, but seems to be the people, or both. Value that, respect that.
As for you worries about challenges/new things/life for her, i have those too with my son. I find though, that the more i respect him for who he is and trust him, the more courageous and willing he is to go out and seek new experiences. I find that he actually is an explorer -- drawn to the unknown, but he definitely has his own pace, which seems to a lot of people "shy" or even "fearful". Those labels have not been helpful to me. I think they are inacurate. Thanks to Elane Aaron's book "the highly sensitive child" i am now on an easier road to loving and accepting him for who he is.
hope this helps. Let us know what happened with gym calss!
best of luck,
D.