[FEATURE] Web UI doesn't update branch name after git branch rename

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

When I asked Claude Code on the web to change what branch it was working on, it was able to change the branch internally and make the git calls it needed to, but the UI on claude.ai/code continued showing the original branch it created for the request.

Proposed Solution

It's a small issue, but it would be nice if the UI updated to the current branch dynamically based on the commands given to Claude throughout the session.

Alternative Solutions

Don't show the branch name in the UI at all, and just make sure you use whatever branch Claude is working in when we click "Create PR"

Priority

Low - Nice to have

Feature Category

Other

Use Case Example

_No response_

Additional Context

_No response_