A longer listing of works by Cooke (though our song is not included) can be found in the Copac academic and national library catalogue:

John Cooke Junior

Since 'Why should London wait?' is there, with a tentative date of 1884, it would seem that the website John quoted earlier was in error in assigning their Cooke (evidently the same Cooke after all) to 'the inter-war period'; the context suggests that they used the term in its usual sense.