I work on the Bitbucket Cloud dev team and am working a similar sounding case with our support team. Looking at the plugin, I can see that it is still using Bitbucket Cloud's very outdated, and deprecated, 1.0 API endpoint for listing branches. The 1.0 API tries to include a diff stat with every branch, causing repositories with a lot of branches & tags (yes, and tags) to hit Bitbucket Cloud's internal timeouts for API performance.
To resolve this issue, the plugin should switch to using the more modern 2.0 API for gathering branch information. The endpoint for getting branch refs is