TortoiseHg is only a Mercurial client, not a Git client. So your Mercurial repository will stay as a Mercurial repository after you move it. GitHub does not support Mercurial, but Bitbucket does.
So you should go create a new Mercurial repository on Bitbucket. They will show you a guide when you create a new empty repository. You basically have to go do the .hg/hgrc
file of the repository and change the default
path so it looks like this:
[paths]
default = https://bitbucket.org/your-username/your-repo
Then push like normal with hg push
or with TortoiseHg. Mercurial will now send all the commits (including all branches) to the repository on Bitbucket.