that'd be Teel-uh-mahns,
short "ee", short 'mute' "uh" and short "ah"
which would make
Teel-uh
eyyyy, Toots is from Flemish Brussels, he speaks Flemish/Dutch, hence his name is pronounced Flemish/Dutch.
Having lived in the US for such a long time, he himself has adapted and when talking english calls himself "Teelmahns", if only just because Charlie Parker did...