There are nonlocal effects in quantum mechanics but I am not sure I would consider quantum teleportation to be one of them. Despite the fact that the dynamics can be nonlocal, it does not therefore follow that all quantum algorithms are actually implementing nonlocality. Some are just local. Quantum teleportation may look at first glance to be nonlocal but it can be trivially fit to local hidden variable models, such as Spekkens’ toy model, which makes it at least seem to me to belong in the class of local algorithms.

You have to remember that what is being “transferred” is a statistical description, not something physically tangible, and only observable in a large sample size. Hence, it would be a strange to think that the qubit is like holding a register of its entire quantum state and then that register is disappearing and reappearing on another qubit. The total information in the quantum state only exists in an ensemble.

In an individual run of the experiment, the qubits may in fact transfer very little information at all between each other, but over the whole ensemble it composes to the quantum state, and thus it then makes sense as to how a joint measurement followed by a transmission of that data over a classical channel could provide sufficient information to move the statistical description of one of the qubits to another entirely locally. Hence, it can be replicated in a local model.