There are two versions of the Burlington Bertie song. The sheet music posted on this page, from 1900, written by Harry Norris and performed by Vesta Shield is for the song "Burlington Bertie". Lyrics are here. Sorry about the unofficial source -- I had trouble finding anything online for this version of the song.
The other, more popular and more comical version is "Burlington Bertie from Bow", which is by William Hargreaves from 1914/1915, and was sung by vaudevillan Ella Shields. It was popular enough in the late 70's/early 80's that some lines about "Lady Diana" were added at the end; in fact I can't find any lyrics online without that bit. Here is a copy of its lyrics.