"her songs were all of champagne suppers and strolling in the Burlington Arcade"

There are two Music Hall songs whose titles include Burlington:

the first is Burlington Bertie, which was composed by Harry B. Norris in 1900 and sung by Vesta Tilley;

the second is Burlington Bertie from Bow (1915), a parody of Burlington Bertie.

The second song is credited to William Hargreaves and was sung by his wife, the male impersonator Ella Shields.

Burlington Arcade is a fashionable shopping arcade in Mayfair, London.