I have to pick up on this since I experienced the issue as well, both on my local machine as well as on the instawp environment Cwicly provides for testing purposes. Latest version of Wordpress and Cwicly are used.
When I try to edit a template (part), changes are not reflected due to API errors. See screenshot attached:
Edit: Cwicly plugin version 1.2.9.9.5 seems fine in this regard. I did some tests on my end and did not come across the issue there.