There is some evidence that the event we now call the Boston Tea Party was not widely known outside of Boston until well after the Revolution. The term "Tea Party" in reference to this event is not known to have existed prior the 1830s, by which point Jacksonian populism had reoriented some of the country's national consciousness toward awareness of the contributions of the "common man." The Boston Tea Party was a subject of a certain amount of awkwardness for some of the Revolutionary movement's more well-heeled, elite management, as they did not like being associated with the destruction of property.

This story is documented in Alfred Young's The Shoemaker and the Tea Party.

As far as what happened in England, the event did help prompt much of the punitive legislation that would become the target of much Revolutionary ire throughout the course of the 1770s, but I am unaware of how well-known it was to the general British public.