My wife's Boustead branch is from Cumberland, England and we have not so far found any who moved from there to London. However, in our researches of the 1851 census for Wigton, Cumberland we came across several Boustead families, some of whom had occupations relating to the fabric trade. For example, Hand loom weavers, dressmakers,tailors and one who was a master tailor employing 2 apprentices and 4 journey men. It may be that there is a link from here.
Also we have identified a William Boustead in my wife's line born Cumberland 1882 although not linked at present to those mentioned in the 1851 census. We have no other information about him but his dates could fit with your grandfather's.
Do you have a date and/or place of birth for your William?