Probably the best single answer is Poincare. Vanden Eynde's history of homotopy introduces the universal cover as a part of Poincare's 1883 work on uniformization, using analytic continuation. ("Development of the concept of homotopy" in I, M, James HISTORY OF TOPOLOGY, p. 82). That would involve both aspects that you asked about.
Vanden Eynde (so far as I can see) does not really say Poincare was the first. Probably it is just too complicated a question when you go into detail.
The themes go back to Abel and Cauchy on Abelian integrals and analytic continuation--in hindsight that was all about connecting paths of integration to series (with radii of convergence) and so to patches covering a domain. It was not clearly understood in terms of covering surfaces until Riemann, and then people took decades to get clear on Riemann surfaces.