Add expiration warnings before cloud session termination

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

Note: This issue assumes the ~7-day session TTL described in #50177 is intentional and documented per #50195. If the TTL is being removed as a bug fix, this issue can be closed.

Cloud sessions on Claude Code on the web (research preview) currently terminate silently at their TTL with no advance warning. When a session expires, it disappears entirely from both claude.ai/code and the Claude mobile app Code tab — not visible even as archived. All uncommitted session context is unrecoverable.

Users have no opportunity to act before their work is lost. There is no banner, notification, or message of any kind before or at the moment of expiration.

Proposed Solution

Surface expiration warnings at multiple time horizons before a session terminates:

1. 72-hour warning — In-session banner: "This session will expire in 3 days. To preserve your work, commit context to GitHub."
2. 24-hour warning — More prominent banner or push notification.
3. 1-hour warning — Alert on next message send: "This session expires in ~1 hour. Save your work now."
4. At expiration — Instead of silent removal, surface a clear message: "Session expired. Here's how to start a new session and restore from GitHub."

Alternative Solutions

Currently there is no workaround — the session disappears without warning and context is unrecoverable. The only partial mitigation is manually committing work to GitHub frequently, but this requires the user to know a TTL exists in the first place (currently undocumented, see #50195).

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

1. I start a cloud session on April 9 and work on a project over several days
2. On April 16 (~7 days later), I open the session to continue work
3. With this feature: I would have seen a banner on April 13 ("expires in 3 days") and again on April 15 ("expires tomorrow"), giving me time to commit my context to GitHub
4. Without this feature (current behavior): the session silently disappears and all context is gone with no recovery path

Additional Context

Related issues:

• #50177 — Bug report (prerequisite: must be closed as "by design" for this FR to be relevant)
• #50195 — Documentation issue (prerequisite: should be addressed first)
• #30922 — Related session lifecycle / orphaned sessions issue