feat(mobile): reuse existing worktrees

[CasperEngl]

What Changed

• Add an existing-worktree picker to the native mobile new-task flow.
• Preserve detached-worktree branch state when creating a thread.
• Add focused coverage for workspace selection validation.

Why

• Mobile needs the same existing-worktree workflow introduced by the parent web/shared PR.

Stack

• Depends on feat(web): reuse existing worktrees pingdotgg/t3code#8517.
• Review and land this together with the parent PR.

UI Changes

• No new screenshots were captured for this split. The code is unchanged from the previously reviewed combined PR.

Checklist

• This PR is small and focused
• I explained what changed and why
• I included before/after screenshots for any UI changes
• I included a video for animation/interaction changes

Implemented with GPT-5.6 Sol through the Codex harness in T3 Code.

[CasperEngl]

Replaced by pingdotgg#8568 so the mobile change is reviewed in the upstream repository.