1) Create a new folder for the fresh repo clones. Navigate inside.
mkdir temp/
cd temp/
2) Clone the repo using the following command.
git clone --mirror git@bitbucket.com:org/name.git
3) Navigate into the locally cloned repo.
cd name.git/
4) Set the new remote
git remote set-url --push origin git@github.com:org/name.git
5) Push the contents of the repo to the new remote.
git push --mirror
Enjoy!