I found this video showing how to review my local changes with Copilot:
https://www.youtube.com/watch?v=qHbVEDNMdGE (The section starts at 5:07)
In the video, you can see there is a button in the source control tab "Copilot Code Review".

I tried to acheive the same but there is no button (with or without Gitlens extension enabled).

I tried to ask in the Copilot chat for a review of my local changes. Then it logged me in to the Github web site and redirected me to a site where I gave permissions to allow access for Code pilot to private and public reposiories. However, the repository I am currently working on is not hosted on Github but on Bitbucket.
Does anybody know how to get this working?
Alright, TechRill youtube channel owner - she is working at GitHub - so few features you can see it may be in private preview and does not applicable for public preview yet. You can also download VSCode Insider and Pre-release GitHub Copilot extension to have the most up to date - just wait for available.
To answer your question, the Code Review feature is scanning local index - so called RAG, and it send the most relevant information to GitHub Copilot backend to handle review, regardless of any kind of Git platform you are using.
Another GitHub product, Copilot Workspace, required GHEC to operate, the mechanism is bigger than Code Review where it will transform your Backlog Story to completely implemented. https://github.blog/news-insights/product-news/github-copilot-workspace/
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