Welcome to Mudcat, Jim!

As for the second part of your query, I remember the Syncopated Lady song and I thought at the time that it was in the style of Fats Waller.

A quick Google shows that the Victoria & Albert Museum, Department of Theatre and Performance, hold a quantity of Ronnie Barker's old scripts, including one called Fats Waller Song, which he wrote under his usual pseudonym of Gerald Wiley.

Script

V & A details

Hope this is useful!

Cheers

LFF