[FEATURE] Desktop app: add /branch and /resume equivalents for managing conversation branches

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

The terminal CLI has /branch (fork the current conversation into a new session at this point, optionally named) and /resume (list and switch back into any prior session, including forks, to continue it). Together they let you manage conversations as a tree: branch off to try something, then come back and resume any branch later.

The Desktop app's Code tab has no equivalent. Typing /branch there returns "isn't available in this environment", and there's no UI action to fork the current conversation's history, nor a way to browse and resume into a specific branch the way /resume does in the CLI. This creates a real capability gap between the two interfaces for the same underlying product.

### Proposed Solution

Bring the CLI's conversation-branching capability to the Desktop app:
- A "Branch conversation" action that forks the current session's history into a new session at that point (optionally named) — the Desktop equivalent of /branch.
- A way to browse and resume any branch (a tree view, or at minimum a flat list showing each branch's origin) — the Desktop equivalent of /resume.

### Alternative Solutions

Currently the only workaround is starting a brand-new session and manually re-typing or re-pasting the relevant context, which loses the actual conversation history and is tedious for anything non-trivial.

### Priority

Medium - Would be very helpful

### Feature Category

Other

### Use Case Example

1. I'm mid-conversation in the Desktop app's Code tab, well into a task with useful accumulated context.
2. I want to try a different implementation approach without risking the current thread.
3. In the CLI I'd run /branch to fork here, try the new approach, and use /resume later to jump back to the original if it doesn't pan out.
4. In the Desktop app I have no way to do this — I either overwrite my current progress or start a fresh session and lose all context.

### Additional Context

_No response_

Start by locating the Desktop app's Code tab and the existing CLI implementations of /branch and /resume. Define how branching, optional naming, and browsing or resuming branches should work in the Desktop app, then verify that a conversation can be forked and that both the original and new branches remain resumable.

Written by the indexing model from the issue text.