Xcode Cloud does not require an SSH key for Bitbucket Cloud. When you click “Grant Access” during setup or under Xcode Cloud → Settings → Repositories → Additional Repositories, an OAuth authorization dialog from Bitbucket will automatically appear.

Simply click “Grant access” in that Bitbucket dialog, after that, the repository is connected to Xcode Cloud, and all builds can access your private Bitbucket repositories.

See https://developer.apple.com/documentation/xcode/connecting-xcode-cloud-to-bitbucket-cloud

Below is an image of the “Grant access” screen in Xcode Cloud (it’s in German, but you’ll get the idea).