This rhyme goes back to at least 1844. The character Punchinello is the same as Punch in Punch and Judy. Punchinello is originally from the 16th-century Italian commedia dell'arte.

Oh, Mother, I'm to Be Married - English Children's Songs - England - Mama Lisa's World: Children's Songs and Rhymes from Around the World  - Intro Image

Notes

Variation:

Oh, Mother, I shall be married to
Mr. Punchinello,
To Mr. Punch,
To Mr. Joe,
To Mr. Nell,
To Mr. Lo,
To Mr. Punchinello.

Thanks and Acknowledgements

This rhyme and illustration can be found in The Sleeping Beauty Picture Book, illustrated by Walter Crane.