Cross-session messaging between Cowork and Code sessions (currently code-to-code only)

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

Cross-session messaging (v2.1.220+) is a genuinely useful handoff channel, but it only spans Code sessions. Cowork sessions are unreachable: tested today on the Windows desktop app (2.1.229) — send_message to a Code session delivered instantly, while sends to two existing Cowork session IDs both returned "Session not found." So the two halves of the same desktop app can't talk to each other.

The practical consequence: a Cowork session that finishes a research or document task has no way to notify or hand off to the Code session that needs the output (or vice versa). Today the handoff is manual — save to a shared folder, then the user re-explains context in the other session. The sessions already share a sidebar and session groups, so the product presents them as one workspace; messaging stopping at the Code/Cowork boundary is surprising.

### Proposed Solution

Extend cross-session messaging across environments: let Cowork sessions appear in the same session directory that send_message/list_sessions use, and deliver messages in both directions (Code -> Cowork and Cowork -> Code) with the same semantics as today — queued until the target's next turn, labeled with the sending session's name, plain text.

Example workflow this unlocks: a Cowork session finishes drafting a report and messages the Code session for the same project "draft saved to /reports, ready for the site build" — the Code session picks it up on its next turn with zero copy-paste from the user.

### Alternative Solutions

Current workaround: the Cowork session writes its output to a shared folder and the user manually opens the Code session and re-explains the context. Works, but loses the notification half entirely — the Code session doesn't know anything happened until the user tells it. Related: #85908 (Desktop Chat conversations similarly cannot be woken by cross-session messages) and closed #42852 (broader multi-surface chat-room request).

### Priority

Nice to have

### Feature Category

Other

### Use Case Example

_No response_

### Additional Context

_No response_

The issue names the send_message and list_sessions entry points; start by tracing how they discover session IDs and deliver queued messages. Done means Cowork and Code sessions appear in the same directory and messages work in both directions with the existing next-turn, sender-label, and plain-text semantics.

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