'Update local repo from remote git repo

I have project on bitbucket, which at its core use another project, that is available on github. I am running old version now of this project, that is on github and I want to update it locally in branch and push it remotely to the BitBucket repo so my coworkers can pull it and also update it. I wonder how to do this with minimun effort. The other project is quite big with a lot of folders and files, I don't want to update it file by file, because it will take me 5 hours. What is a clever way to pull the changes from the github repo to my local branch and then add, commit and push to BitBucket?



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