[FEATURE] Show pending update notification on session start when native installer has downloaded a newer version

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

The native installer downloads Claude Code updates in the background, but there is no indication to the user that a newer version is available until they manually exit and re-enter the CLI, or run claude --version / claude upgrade on their own.
I was running v2.1.94 for several sessions without realizing v2.1.101 was already downloaded and waiting. The only reason I discovered this was by manually checking the version against the changelog. This means users can run stale versions for days or weeks; missing bug fixes, security patches, and new features; with zero visibility into the fact that an update is sitting there ready to apply.
This is especially relevant for long-running sessions using --dangerously-skip-permissions or Agent Teams, where sessions can last hours and users rarely restart the CLI.

Proposed Solution

On session start, if the native auto-updater has already downloaded a newer version than the currently running binary, display a single non-blocking line before the prompt:
Update available: v2.1.101 (restart Claude Code to apply)
No action required from the user; purely informational. It should not interrupt the session flow or require dismissal. Similar to how brew shows "X outdated formulae" or how npm prints update notices.

Alternative Solutions

Currently the only workarounds are:

Manually running claude --version and comparing against the changelog
Running claude upgrade periodically to check
Exiting and re-entering the CLI on a regular cadence and hoping the version number changes

None of these are discoverable or natural parts of a coding workflow.

Priority

Low - Nice to have

Feature Category

CLI commands and flags

Use Case Example

I finish a long Claude Code session at 11pm and close the terminal
Overnight, the native installer downloads v2.1.101 in the background
Next morning I open Claude Code; it launches v2.1.94 (the old binary still in memory/path)
I work for 3 hours without knowing a newer version exists
I happen to run claude --version, notice I'm behind, exit, re-enter, and now get v2.1.101

With this feature, step 3 would show "Update available: v2.1.101 (restart Claude Code to apply)" and I would have immediately restarted before beginning my work.

Additional Context

This is analogous to how VS Code shows "Restart to Update" in the status bar, or how Homebrew prints outdated formula counts. The native installer already does the hard part (downloading the update); this just closes the loop by telling the user about it.