He was best known for his work with the Bonzo Dog Doo-Dah Band, The Rutles and collaborating with Monty Python.
A spokesman for the Innes family said he had not been suffering from any illness and had passed away unexpectedly on Sunday night.
Innes had been travelling home from France with his family, who have asked for privacy “at this difficult time”.
“It is with deep sorrow and great sadness that we have to announce the death of Neil James Innes on 29 December 2019,” they said in a statement.
“We have lost a beautiful, kind, gentle soul whose music and songs touched the heart of everyone and whose intellect and search for truth inspired us all.
“He died of natural causes quickly without warning and, I think, without pain.”
It went on: “His wife Yvonne and their three sons, Miles, Luke and Barney, and three grandchildren, Max, Issy and Zac, give thanks for his life, for his music and for the joy he gave us all.”