Hi, Thanks for the detailed background info.
There is a Chates Farm north of Henfield, just before the Cowfold intersection, but more probably, the birthplace you want is around Chates Cottage also in that area. Perhaps this was a place chosen at random for the birth, but it might be worth investigating the 1901-1911 censuses to see if any distant relative lived in that locality where discrete births could be conducted.
You made no comment about those on Genes that have your Cicely on their tree, who clearly knew of her but perhaps not of her indiscretion when she was about 16. Anyway, I highly reccommend you get onto the Genes site and try to make contact with one or all of them.
You will also find a number of Bedwell messages amongst earlier Rootschat correspondence, and a Bedwell researcher within the membership of The Sussex Family History Group.
Having checked your other messages on the Cumberland Rootschat website I discovered an interesting side issue which may or may not add something to your research. Although Manchester (where William died) is a large city, there was only one Bedwell entry amongst the births. Hilda Bedwell, mothers name Williams, born 1929. With no local Bedwell-Williams marriage to back it up. I wonder whether your William's absence was because he had now committed an indiscretion of his own?
Hope this helps you further.
Roy G