For the uninitiated, like me ten minutes ago, this is the uniform under discussion.

There does not seem to be a clear answer. Elizabeth Hennessy, in A Domestic History of the Bank of England, states that this is worn by about a third of the parlor messengers of the bank (the most senior? She doesn't say), and that it may have been the livery of the servants of John Houblon, the first governor of the bank, ca. 1694.

Is this plausible? Livery - clothing for male servants in the colors of the household - was in use at this time. Blue was a very common color, but properly a family's livery would either match its coat of arms (The Wordsworth Dictionary of Phrase and Fable, 2001) or be a specific color chosen to be used by the family (Lucy Worsley, Cavalier, 2011 - she points out that the Cavendish livery was dark green, and the Cavendish crest is black and white). Technically the Houblons were not ennobled, so a shield will not help; I can't find any references connecting the Houblons to pink, either.

I have to note that I can't seem to find a reference to this livery prior to the late 19th century (where it's presented with no history), but it's possible that they exist.