He died suddenly while at home with his wife on Saturday, his agent said.
The Liverpool-born actor lent his voice to children's favourite Thomas and Friends for 13 series after taking over from Beatle star Ringo Starr in 1991.
He also appeared as Mickey Startup in Auf Wiedersehen, Pet.
He starred alongside Bernard Hill and Julie Walters in the Bafta-winning 1980s series Boys From The Blackstuff, which highlighted the hardships of unemployment.
Angelis also had roles in Minder, The Liver Birds, Z Cars and Good Cop.
He was married to Coronation Street actress Helen Worth, who plays Gail Platt, from 1991 to 2001, and later married Jennifer Khalastchi.
Paying tribute to him on social media, comedian Matt Lucas described Angelis as "one of the greatest TV actors I've ever seen".
Lucas said: "His work with (screenwriter) Alan Bleasdale was tremendous. What a loss."