Firstly, we have already addressed the question of what licence applies to GitHub-hosted content in the absence of a specific licence declaration, and the prevailing (though not the only) opinion seems to be that
if you publish your source code in a public repository on GitHub, you have accepted the Terms of Service which do allow other GitHub users some rights. Specifically, you allow others to view and fork your repository.
When you say "fully copyrighted", I presume you mean "all rights reserved" (even GPL'ed code is fully-copyrighted, that's why the GPL works). But it's clear that some rights are given away by the mere act of publishing on GitHub. In that sense, GitHub is not a very good backup medium, and you should make better arrangements immediately.
Is that enough to comply with the GPL? You don't say which version applied to the code you're working on, but let's assume it's v3. Section 5(c) of GPLv3 says that you may convey modified source versions but
You must license the entire work, as a whole, under this License to anyone who comes into possession of a copy.
It also says that
To “convey” a work means any kind of propagation that enables other parties to make or receive copies.
That given, and noting that IANAL/IANYL, putting a work on GitHub looks to me like conveying it, which means that you must license the work under GPLv3 (and comply with its other obligations). If you do not convey your modified code, and instead you keep copies only on your own computer(s), and until you do convey it to some other party you may keep and tinker with it to your heart's content without any further obligations. Similar arguments apply if the original code was licensed under GPLv2.
