Using Bitbucket, how do I get commits from one branch into another?

You cant, you'll have to use git commands. Would have been nice with a web interface for it like pull requests.

Use git cherry pick. It lets you pick commit and add them over other branches.

git cherry-pick <commit> 

Example:

git cherry-pick 123456

This will add the commit on the current branch.

As Tim suggested, you'll want to merge new-features into fix-these.

You can do this on Bitbucket by navigating to the Compare page (see sidebar), selecting "fix-these" as destination, "new-features" as source and then hitting the Merge button in the top-right corner.