The "escaped Nazi" narrative comes from a documentary of SS-Hauptsturmführer and Gestapo member Klaus Barbie directed by Kevin Macdonald titled My Enemy's Enemy. The documentary alleges Klaus Barbie helped the CIA capture Che Guevara. It was specifically journalist Kai Hermann who made this claim, saying "[Barbie] always boasted – though I cannot prove it – that it was he who devised the strategy for murdering Che Guevara" as well as a friend of Barbie, Alvaro De Castro, saying "He (Barbie) met Major Shelton, the commander of the unit from the US. (Barbie) no doubt gave him advice on how to fight this guerrilla war. He used the expertise gained doing this kind of work in World War Two. They made the most of the fact that he had this experience". This "guerrilla war" refers to the Guevara-led ELN (Ejército de Liberación Nacional de Bolivia), a group of mostly Bolivian and Cuban guerrillas, and their revolutionary goal to overthrow the conservative President René Barrientos Ortuño (a military officer who came to power after a CIA-backed coup overthrew the democratically elected President Victor Paz Estenssoro).

On October 7th, 1967, Guevara was wounded and taken to the village of La Higuera as prisoner. He was very defiant, reportedly kicking a captor (Bolivian Officer Espinosa) who came to take the pipe from his mouth. The Bolivian President René Barrientos, despite the US wanting Che Guevara to be transferred to Panama for interrogation, ordered Guevara's death. However, the government didn't want Che's death to seem like an execution, so, the chosen person to carry out the execution (Bolivian Sergeant Mario Terán) was ordered to not shoot Guevara in the head. Mario Terán entered the hut in which Guevara was being held, and ordered everyone else out. Che Guevara then said "I know you've come to kill me." Mario Terán then, hesitatingly, pointed his M1 Garand at Che Guevara to which Che replied "Shoot, coward! You are only going to kill a man!" He was then shot several times, becoming deceased.

Regarding Klaus Barbie, it's true he was employed by the US (for intelligence) who helped him escape to South America, and it's also true had lived in Bolivia (as Klaus Altmann) and had a close relationship with Bolivian officials but the extent of Klaus Barbie's participation in the hunt for Guevara remains disputed.

Che Guevara was wounded and captured during an armed encounter between the guerrilla forces he led and the Bolivian army. After being captured, he was interrogated inside a nearby schoolhouse and then executed outside. He was shot by a Bolivian army sergeant named Mario Terán. The order came from the Bolivian president. A CIA agent (Félix Rodríguez) was present on scene, but I've come across nothing regarding the involvement of escaped Nazis.

Find a copy of Jon Lee Anderson's Che and read Chapter 29 if you'd like to know more.