There are numerous English language schools which are looking for native speakers as teachers. They can arrange papers or pay teachers under the table. The pay is not good, but enough to live on in Montevideo. Work permits are not particularly difficult to obtain and Uruguay lets you convert a tourist visa to a work visa without leaving the country. Residency visas without permission to work simply require you prove access to $500 USD a month. Work permits are not particularly hard to get.
One of the best experiences to have while your stay at Uruguay is to watch a game between Nacional and Peñarol, the two most followed football teams in the nation.