The British would hire transport ships to move the bulk of their troops, not Royal Navy Vessels. For amphibious operations, the transports would carry flat bottom barges. As part of Amherst's expedition against Ticonderoga in 1759, there was at least 1 flat bottom "barge" - 36 foot long, 14 oar and 50 men. So besides Quebec and Louisbourg and the Caribbean Campaigns, this barge is a very good fit for Ticonderoga. Amherst also inherited a 100-ton sloop from Abercromby. To ensure no interference from the French, an artillery barge (sail/row) mounting seven 24-pound cannon supported the landing at the north end of Lake George. Amherst also had a flat bottom transport capable of transporting 70 horses.
By the end of 1756, a British "navy" on Lake George had been established at Fort William Henry. The British had two small 20-ton sloops armed, one armed with 4 swivels and the other with 2 swivels, and a 30-ton sloop armed with two 6-pounders, one 7-inch mortar and 8 swivel guns (Earl of Loudoun). An even larger 40-ton sloop (George ?) was in stocks under construction. By Fall 1756, the 3-sloop force was now capable of conducting patrols over the entire length of Lake George. The French had no boats capable of opposing them on the Lake. In addition, the British had three scows, flat bottomed sailing vessels that may have been designed to transport artillery for the Winslow's aborted attack on Carillon. Alternatively the three scows were "gundolas", a sail/row vessel mounting a single cannon and a pair of swivel guns.
In March 1757, the French attacked Fort William Henry. The two larger sloops were burnt, total losses. The two smaller sloops were damaged by fire, but were repaired. Apparently, the storehouses holding the rigging for the ships were also burned so the sloops were still "docked" and unavailable to oppose Montcalm in August of 1757. The "gundolas" suffered from a similar lack of rigging. Potential for Snowshoes Figures here as this attack on Fort William Henry occurred in the deep snow.
In Summer 1757, the British planned to attack Canada out of Halifax, Nova Scotia. The French Navy prevented the British attack. But as the British were planning to invade Quebec via Nova Scotia, the thought was Montcalm would be stuck on the defensive all summer in Canada. New York was expected to be very quiet in Summer 1757, so there was no rush to repair the British Navy at Fort William Henry and no New York based expedition was planned -- just guard duty. With the British beached in Nova Scotia (Including Major Roberts), Montcalm was free to attack the British at Fort William Henry and captured the fort in August 1757.