In Timeline from company-managed boards, it is possible to display a timeline bar for tasks when they belong to a Sprint with start and end dates but cross-board Sprints are not supported so if the Sprint was created in a different board the bar will not be displayed.

That's also the case when you move issues between projects. In such scenarios the sprints of the source project will still be tied to their board of origin and will not be displayed in the new project's Timeline.

Use the next REST API call to get the origin board.
https://<site>.atlassian.net/rest/agile/1.0/sprint/<sprintId>