Andy,
Sorry for the delay in replying. As far as I am aware the William Coates & Co, 25 High St, Whitechapel which opened about 1820 and 18 Adam St, Adelphi were wine merchants which was before the time frame you quoted. There are records of this firm held at The Guildhall Library.
Could there have been another firm of the same name, or possibly a later generation of the family?
I would be interested to hear if this is the case.
Best Wishes
Scatty