If you are looking for Annie Smith born to an Annie Smith, possibly illegitimate, one you may come across is the Annie Elizabeth Smith born 29 Aug 1901 in Lambeth Workhouse. Baptism same day.
Unfortunately the mother looks to be a few years older than Annie Fowler, and apparently was Annie Maria.
I don't know what happened to Annie Fowler. If that is them in Harrow in 1939, they were still there to about 1952 I think. Then I lose them, with possibly Walter dying in Camberwell later. All a bit messy
Two things about the illegitimate birth in Lambeth. Mother Annie may have been born in November, but no record that I can find names a relative for her. She was apparently a cook, and there is a settlement exam on ancestry. Addresses she has been at are given, but she is not at the one she might be expected to be found in the 1901 census.
A coincidence is that the cause of her trouble, as she puts it, meaning the putative father of her child, was Walter Batchelor from 13 Railton Road, Herne Hill. A coincidence certainly, but probably only that.
His father seems to be William (says Walter here)
https://www.ancestry.co.uk/imageviewer/collections/1557/images/31952_a013090-00217Another record here
https://www.ancestry.co.uk/imageviewer/collections/1557/images/31363_a109806-00048There was a removal order to Camberwell
https://www.ancestry.co.uk/imageviewer/collections/1557/images/31436_190169-01070Just so you have some more info on this one, if you can get hold of those images. I am not saying she is the right one!