I'm not sure what you're getting at here. Even under Copenhagen, one can duplicate an upload as it's running.

Let's suppose that your mind is a function of your brain, and that your brain is composed of atoms.

In MWI there are many branches with many configurations of atoms, that means many branches of your brain, that means many branches of your mind. In every branch your mind is entangled with the other atoms of the same branch. So for example in the universe with atoms of a dead cat, your mind is in the "poor kitty" state, and in the branch with the atoms of an alive cat, your mind is in the "kitty, you are so lucky, I promise I will never try this cruel experiment on you again" state.

In Copenhagen, on a tiny time scale there are many branches of atoms. But it is believed that on a larger scale it is not so. At some unspecified moment there is supposed to be a collapse where many branches of atoms become a single branch again (through a process of random selection). Nobody knows when does this happen. On a large scales, we are not able to run a precise enough experiment that would say either way. On smaller scales, where we can run the experiment, the result has always been that the collapse did not occur yet. So after the collapse, there is only one branch, and therefore one mind. Before the collapse... I would say that there is a superposition of minds (because there is a superposition of brains, because there is a superposition of atoms the brain is composed of), which should become one mind again at the moment of the collapse. But it is believed that this superposition exists only for a very small fraction of the second, so it's not like the different minds in the superposition have enough time to really think significantly different thoughts. The neurons work at a limited speed, and sending a signal from one neuron to another requires dozens of chemical reactions.