R' Nachman's actual statement from Likutei Moharan II 48 is:
וְדַע, שֶׁהָאָדָם צָרִיך לַעֲבר עַל גֶּשֶׁר צַר מְאד מְאד וְהַכְּלָל וְהָעִקָּר שֶׁלּא יִתְפַּחֵד כְּלָל
"And know, a person needs to make his passage on a very, very narrow bridge, and the rule and the essence is to not be afraid at all." (*)
Likutei Moharan II 24:
מִצְוָה גְּדוֹלָה לִהְיוֹת בְּשִׂמְחָה תָּמִיד
(*) Regarding whether this is misquote: The context there is a person must be very persistent and get through the obstacles he is usually shown (outwardly, from Heaven, though H"Y truly and greatly desires our devotion) when he tries to enter the service of Hashem. In the sense that the Hebrew word for world, עולם *OLaM*, relates to concealment, העלם *heELeM*, this song adaptation does convey the original to some degree.
The popular song adaptation in Hebrew, כל העולם כולו גשר צר מאוד והעיקר לא לפחד כלל, was composed by Boruch Chait of The Rabbi's Sons.
It has since been variously rendered in English as "The whole entire world..." etc. (in order to fit the notes of that melody).