This was a well documented event. There is even an engraving which shows the Balcony where the Royal Family stayed to watch the procession.
Here is a, roughly contemporary, picture showing Bow Church on a less crowded day.
And I have found a further account of the event.
"Sir Samuel Fludyer, Lord Mayor of London in the year 1761, the year of the marriage of good King George III., appears to have done things with thoroughness. In a contemporary chronicle we find a very sprightly narrative of Sir Samuel's Lord Mayor's show, in which the king and queen, with "the rest of the royal family," participated—their Majesties, indeed, not getting home from the Guildhall ball until two in the morning. Our sight-seer was an early riser. He found the morning foggy, as is common to this day in London about the 9th of November, but soon the fog cleared away, and the day was brilliantly fine—an exception, he notes, to what had already, in his time, become proverbial that the Lord Mayor's day is almost invariably a bad one. He took boat on the Thames, that he might accompany the procession of state barges on their way to Westminster. He reports "the silent highway" as being quite covered with boats and gilded barges. The barge of the Skinners' Company was distinguished by the outlandish dresses of strange-spotted skins and painted hides worn by the rowers. The barge belonging to the Stationers' Company, after having passed through one of the narrow arches of Westminster Bridge, and tacked about to do honour to the Lord Mayor's landing, touched at Lambeth and took on board, from the archbishop's palace, a hamper of claret—the annual tribute of theology to learning. The tipple must have been good, for our chronicler tells us that it was "constantly reserved for the future regalement of the master, wardens, and court of assistants, and not suffered to be shared by the common crew of liverymen." He did not care to witness the familiar ceremony of swearing in the Lord Mayor in Westminster Hall, but made the best of his way to the Temple Stairs, where it was the custom of the Lord Mayor to land on the conclusion of the aquatic portion of the pageant. There he found some of the City companies already landed, and drawn up in order in Temple Lane, between two rows of the train-bands, "who kept excellent discipline." Other of the companies were wiser in their generation; they did not land prematurely to cool their heels in Temple Lane, while the royal procession was passing along the Strand, but remained on board their barges regaling themselves comfortably. The Lord Mayor encountered good Samaritans in the shape of the master and benchers of the Temple, who invited him to come on shore and lunch with them in the Temple Hall.
Every house from Temple Bar to Guildhall was crowded from top to bottom, and many had scaffoldings besides; carpets and rich hangings were hung out on the fronts all the way along; and our friend notes that the citizens were not mercenary, but "generously accommodated their friends and customers gratis, and entertained them in the most elegant manner, so that though their shops were shut, they might be said to have kept open house."
This is taken from Walter Thornbury's Book of Old and New London pub 1887 but still in print!
The account goes on to tell us that Thomas's "Quaker" was a Mr Barclay.
Samuel Fludyer seems to have been the 18th century equivalent of camera shy (I don't believe it. I'm sure there is a portrait somewhere). However the elaborate wig he wore for his Lord Mayor's Show is depicted in William Hogarth's Five Orders of Periwigs. It is the fancy one on the far right with all those "wings".
Sir Samuel would have travelled in the the same Lord Mayor's State Coach as is used today. It was made in 1757.
It is now in the Museum of London, when the Lord Mayor doesn't require it. You can visit it.