I suppose that Borel is mentioned in the text you refer to by mistake. I cannot prove this (Borel has 335 publications including 85 books according to Zentralblatt) but a large recent book on the uniformization theorem and its history

H. P. Saint-Gervais, Uniformisation des surfaces de Riemann,
ENS Editions, 2010,

does not mention Borel. (And I've never seen Borel mentioned in connection with the uniformization theorem).

The proof based on the equation
$$\Delta u=e^u$$ is due to Picard (the first paper he wrote on this was in 1890). Then he wrote many subsequent papers commenting on and improving this result. All these papers can be easily found in Zentralblatt since they all have the  equation mentioned above in the title. His final publication on the topic was a chapter in his book

E. Picard, Quelques applications analytiques de la theorie de courbes algebriques, Gauthier-Villars, 1931.