There are actually three different clauses at work here:
大政治家になりたい = "I want to become a great politician"
大望を忘れた = "He forgot the ambition"
・・・ことはなかった。= "It was never the case that [...]"
The first word you have highlighted, という, serves to show the relationship between the first two clauses. It is literally と "with" plus いう "X says". His ambition can be stated to be, "I want to become a politician."
The second word you highlighted, なかった, is the end of the operating clause of the sentence. It's the past tense of ない. Something did not exist. What does not exist? An instance, こと。 What instance did not exist? The instance of forgetting ambition.
So, as for him, there was not an instance where he forgot the ambition which could be stated as, "I want to become a great politician." The translation you have given phrases this more clearly.