Elizabeth Watson was born about 1793 at Winchelsea, Sussex, England. She is listed as mother to William Bourne Russell Watson, born 30 June 1815 at Winchelsea, Sussex, England. Family folk lore says she did not marry the father but that they remained friends. I can find no marriage or death record in England for Elizabeth. What other records are available to help find what may have happened to Elizabeth?
In the 1861 Census of England and Wales, there are more than 50 women named Elizabeth Watson born in the 18th century. There were over 100 in the 1841 Census.
If your target person did not marry or die (and you say that you have no evidence for either of these), then we can assume that she is one of those people (or emigrated, or changed her name, or ...).
Your problem now becomes, what do you know about your Elizabeth that will let you recognise her in the census records. Obviously, her age and place of birth are starting points.
It's true that she never married the father of WBR Watson - indeed she had to chase him for child maintenance! There are a range of 'bastardy' documents in the Sussex archives (search at the keep) where the local parish authorities had an arrest warrant made out to detain Bourn Russell so he would pay for the upkeep of the young WBRW. Doesn't sound too friendly,! But old Captain Russell took young WBRW on as a ship's boy, so maybe they worked it out in the end.