Tiles in the Background Tasks tab should show start/finish timestamps

### 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

When I view the Background Tasks tab in the Claude Code desktop app for macOS, some tiles indicate errors that happened hours ago: there is no way to tell whether the error is fresh or not. So it would be helpful if each tile had a timestamp.

### Proposed Solution

It would be helpful if each tile had a timestamp.

### Alternative Solutions

_No response_

### Priority

Medium - Would be very helpful

### Feature Category

Interactive mode (TUI)

### Use Case Example

_No response_

### Additional Context

_No response_

Locate the Background Tasks tab and the code that renders its tiles. First determine how each task records its start and finish times, then display those timestamps on the tile. Done means the timestamps are visible for the relevant task states and remain useful for distinguishing recent errors from older ones.

Written by the indexing model from the issue text.