Remote Control: show nested (child) sessions on GitHub Mobile and github.com, like the VS Code Agents window #5316

Describe the feature or problem you'd like to solve

In the VS Code Agents window, a session can spawn child sessions (nested sessions / sub-chats). The sidebar shows them as a tree under the parent, for example Playground Project → Greeting, and I can open each one and read or steer it.

When I steer the same parent session from GitHub Mobile or the github.com task page via /remote on, the child sessions are not visible at all. I only see the parent conversation. Work that the parent delegated to a child (and the child's questions, permission prompts and results) is invisible from the phone, so I can't follow or steer it until I'm back at the desk.

Proposed solution

Mirror the Agents window's session tree in Remote Control:

• List child sessions under their parent (with status: running, waiting for input, done) on GitHub Mobile and github.com.
• Open a child session to read its transcript and steer it, with the same Remote Control semantics as the parent.
• Surface a child's pending permission prompts or questions in the parent's view (or as a notification), so a blocked child doesn't stall silently.

Example prompts or workflows

1. In VS Code, ask a session to "create a Greeting chat for this task and hand it the details" (nested session).
2. Leave the desk, open the session on the phone: the sidebar/tree shows Greeting under the parent with its status.
3. Tap Greeting, read what it did, answer its question by steering it, go back to the parent.

Additional context

• The Agents window already models this (parent/child sessions, "Open in Agents window", nested sessions with separate workspaces as of VS Code 1.144), so the data exists host-side; this is about exposing it through Mission Control / Remote Control clients.
• Related Remote Control work: agentHost: render Remote Control prompts as user turns microsoft/vscode#345426 (remote prompts rendering), Remote Control: voice mode from GitHub Mobile (talk to the session, hear spoken replies) #5315 (voice mode for Remote Control).