[Feat] Add live nested task views to web sessions

[roomote-roomote]

​Created by Roomote. Follow up by mentioning @roomote-roomote, in the web UI, or in Slack.

What changed

• Render successful delegated task launches in web session transcripts as live task-aware cards with current task status.
• Open delegated task details in the session's existing resizable side-panel slot, showing the child transcript without nested task navigation or action-rail chrome.
• Add a clear Go to task action that opens the full child task UI.
• Preserve task cards as standalone transcript entries and reuse the existing task polling, history, and sandbox realtime providers for updates.

Why this change was made

Web sessions previously exposed delegated launches as generic tool activity, forcing users to leave the parent conversation to understand child progress. This gives nested work a focused live view while keeping the established panel and task UI conventions.

Impact

Users can monitor and inspect delegated work without losing the parent session context, then move into the full child task when they need its complete controls. Focused web tests, type checking, ESLint, and the repository pre-push checks pass. Browser screenshots could not be captured because the local environment redirects Sessions into incomplete onboarding and could not finish inference-provider setup.

[roomote-community]

No code issues found. See task

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