It was common to take your house name with you when you moved. The Simons did live at a Gladwyn in Cromwell Avenue, Bromley. Kelly's 1913 directory has:-
Numbers 1,2,3,5,6 then:
Bromley Cottage Hospital
... here is footpath to Hayes ...
Simon Charles Victor, Gladwyn
Bodle E. W. C., Somerton
Ashton Pearson, Whitefriars
Maggs P. H., Belstone
Hale George Herbert, Tressillian
The latter houses weren't in the 1910 directory. They were still there in 1916, but as you know, gone by 1924, the next directory I have.
Without access to the full run of directories at Bromley Local Studies, I can't be certain, but following through those I have here, I think it's now No 25 or 27 in the row beginning with 21 after Cromwell Close to the end of the Avenue.