Create Web IDE MR and branch picker

What does this MR do?

This MR removes the existing MR picker in the Web IDE, and replaces it with a new branch / MR picker.

Notes:

• This points to !20850 (merged) (which addresses #47768 (closed)) and contains the general Web IDE header redesign.
• The plan is to approve the two parts separately, then merge this into the single feature branch, and then into master.

Are there points in the code the reviewer needs to double check?

Why was this MR needed?

• #46165 (closed)
• #47768 (closed)

Screenshots (if relevant)

Does this MR meet the acceptance criteria?

• Changelog entry added, if necessary
• Documentation created/updated
• [ ] API support added
• Tests added for this feature/bug
• Conforms to the code review guidelines
  • Has been reviewed by a UX Designer
  • Has been reviewed by a Frontend maintainer
  • Has been reviewed by a Backend maintainer
  • [ ] Has been reviewed by a Database specialist
• Conforms to the merge request performance guidelines
• Conforms to the style guides
• Conforms to the database guides
• If you have multiple commits, please combine them into a few logically organized commits by squashing them
• Internationalization required/considered
• End-to-end tests pass (package-and-qa manual pipeline job)

What are the relevant issue numbers?

Closes #46165 (closed)