The British Library Archival Sound Recordings has this brief excerpt:

Oh, I don't need a knocker-up; I don't need a clock,
For underneath the bed I pop the old game cock.
Every morning, every morning, I never oversleep, it's true,
For out I go when the cock begin to crow,
With 'is cocky-ocky-ocky-doodle-doo.


And here is the description from the catalogue:

Duration: 00:00:36
Shelf mark: 1CDR0010662 (copy of C1002/89)
Subjects: English folk songs and music (Suffolk); Music hall
Recording date: 1973
Collection: Keith Summers Collection
Recording locations: Worlingworth Swan, Worlingworth, Suffolk, England, UK
Performers: Arthur
Recordists: Summers, Keith
Description:
Item notes: First line: Oh I don't need a knocker up, I don't need a clock. Short ditty. Possibly the same song as 'Every morning' (Roud Folk Song Index No. 10733). Recording notes: Mono

They give it the title COCKY-OCKY-OCKY-DOODLE-DOO but I suspect that is arbitrary.