On the events of the 16th of December 1773 in Boston Harbor, the Boston Tea Party Historical Society article British View vs. American View, states that the incident was
Scarcely noted in the British press at first
when the news reached London on the 20th of January 1774. Thus, the ‘average British person’ would probably not even have known about it when it was first reported. A copy of an article from the December 20th 1773 issue of the Boston Gazette was copied by the London Chronicle dated January 22nd 1774. In part, it reads:
A number of brave & resolute men, determined to do all in their power
to save their country from the ruin which their enemies had plotted,
in less than four hours, emptied every chest of tea on board the three
ships commanded by the captains Hall, Bruce, and Coffin, amounting to
342 chests, into the sea!! without the least damage done to the ships
or any other property. The matters and owners are well pleas'd that
their ships are thus clear'd; and the people are almost universally
congratulating each other on this happy event.
Even if the 'average British person' did see or hear about this report, the significance of the Boston Tea Party would not have been immediately evident to most people and would not have appeared to be as serious as the 1770 Boston Massacre which involved loss of life, or perhaps even the Gaspee Affair of 1772 when a Royal Navy ship was looted and burned.
This, as already noted by the OP, was not the case with Parliament, where the majority of MPs’ reaction was to support stern measures against the Bostonians. Parliament, though, was concerned with protecting its own authority and was not elected by a representative cross-section of the British population, voters being males over 21 who met minimum property requirements. In the election of October – November 1774, even this limited electorate (unlike those they elected) had other priorities:
The problem of colonial policy played little part in the elections (it
was an issue in only ten of the 314 constituencies)
Source: B. Knollenberg, Growth of the American Revolution
There was, however, another group who were aware of the Boston Tea Party – merchants. Their concern was, of course, self-interest; any disruption to trade would cost them dearly, but it is difficult to separate reactions to the Boston Tea Party from other events (such as the Intolerable Acts) in the wake of December 1773. Nonetheless, in British Supporters of the American Revolution, 1775 - 1783, Sheldon A. Cohen notes that,
After the Boston Tea Party in 1773, several London merchants held out
the vain hope that some form of reconciliation could be reached with
the North American colonists.
London merchants were not the only ones who were both aware and concerned. Cohen again:
Portsmouth was not oblivious to the crucial developments in the North
American settlements. The town regularly received London newspapers
and other publications featuring stories about American unrest. In
addition, since it was a seaport, Portsmouth’s merchants were well
aware of colonial boycotts of British goods and of the destruction of
East India tea in Boston Harbor in 1773.
Among the cartoons and prints relating to the Boston Tea party is this British one Published by Sayer & Bennett (London, October 31, 1774) (Robert Sayer and John Bennett were Fleet St. publishers):
“A British view in 1774 of how authority was being challenged in Massachusetts. Before a “liberty tree” on which the Stamp Act placard is tacked upside down, a tarred-and-feathered Tory is being forced to drink taxed tea while in the harbour the Boston Tea Party goes on. A gibbet hangs from the liberty tree.” Source: Stanley Weintraub, Iron Tears
The above image also illustrates how the Boston Tea Party was commented on along with other issues happening around the time, rather than in isolation. The tarring refers to the aftermath of an altercation involving a British customs officer, John Malcolm, in January 1774.
This next British print makes no direct mention of the Tea Party but is clearly a result of it:
Print shows Bostonians held captive in a cage suspended from the "Liberty Tree." Three British sailors standing in a boat feed them fish from a basket labeled "To -- from the Committee of --" in return for a bundle of papers labeled "Promises"; around the tree and in the background are cannons and British troops. The paper in the hand of one jailed Bostonian says "They tried with the Lord in their Troubl & he saved them out of their Distress. CVIL 13." London : Printed for R. Sayer and J. Bennett, Map & Printsellers, No. 53 Fleet Street, as the Act directs, 1774 Novr. 19.
Notable individuals outside of Parliament also had something to say on the general situation in America:
In Taxation No Tyranny, published in 1774, Samuel Johnson, the
compiler of the first modern English dictionary and an influential
British writer, set the tone, asking readers how it was “that we hear
the loudest yelps for liberty among the drivers of negroes.”
Finally, it should not be forgotten that not all Americans approved of what happened in Boston. Benjamin Franklin, who was in London at the time, and George Washington both condemned the destruction of property and wrote that all the destroyed tea should be paid for.
Note on the term 'Boston Tea Party'
The event wasn’t even labeled as the Boston Tea Party until 1826, and
at this time the term “party” was used to refer to the men involved,
not the event itself (they were “of the Tea Party,” not “at the Tea
The Electoral System and Parliamentary Reform (PowerPoint)
How Did the British Press Cover the American Revolution?
Lewis Namier, John Brooke, The House of Commons 1754-1790