So really this question is something of a misnomer, there were very few British attempts to reach the North Pole via ship prior to the 20th century. Most early naval expeditions by the British were searching for the Northwest passage to Pacific, rather than trying to reach the Pole. Some 19th century missions (such as the infamous Franklin expedition) tried to reached the magnetic north pole, which at the time was located in the modern day Canadian Arctic.

The only two British naval expeditions with the stated purpose of reaching the North Pole of the 19th century was the Parry expedition of 1827 and the British Arctic Exhibition of 1875. Parry's expedition launched from Svalbard, so they would have never encountered any circumpolar peoples. The BAE went up the Smith Sound between Greenland and Ellesmere Island, they would encounter the Inuit, but by this point British/Canadian explorers would likely not be too much of a surprise to them.

Remember, British ships had been showing up in the Arctic for 300 years by this point, with Martin Frobisher making contact with the Inuit on his first voyage in 1576. The Inuit were a highly mobile people and word almost certainly got around, even the most insular of Inuit in the 1870s probably knew about the the Kabloonas, their big ships, and their rectangular tupiks.

The same would also be true of the British explorers, they knew about the Inuit and expected to meet them. The BAE even had two Greenlander Inuit guides onboard. This might be somewhat different with earlier exhibitions, but even the earlier exhibitions to the Arctic of the 1820's - 1840's (Ross, Parry, Franklin etc) were expecting to encounter Inuit. Franklin himself had been the Arctic twice before, and partially spoke Inuktitut.

How the Inuit saw the explorers is a much more complicated matter, because of how differently their encounters could be. Frobisher's initial hostile contact of skirmishes and kidnappings in Hudson bay were extremely different to what the Inuit encountered with Ross (mostly peaceful trading with the odd bit of thievery) and the Franklin expedition: hundreds of the starving, frostbitten, dying men who left a trail of bodies down the coast of the coast of King William's Island during Crozier's doomed march south.

But the Franklin expedition was really when westerners became interested in what the Inuit had to say about the explorers, as they were the only living witnesses to the fate of Franklin and his men and so those searching for the truth took to interviewing them. Hence these disturbing encounters live large in the surviving oral testimony of the encounters between Inuit and the British explorers.

You can find a collection of these oral testimonies here: https://www.canadianmysteries.ca/sites/franklin/archive/archiveAudioInuitTestimonyIndex_en.htm

As for the furthest North that you find Inuit, the Inughuit, of North Western Greenland were contacted in 1818 on John Ross's first Artic Expedition. I do not know if Sir John was surprised to encounter them, but no latter exhibition would be surprised by anyone living further North, because there were none. Many later exhibitions of the 19th century did not even try to venture so far north. Much of the later exploration was about mapping the interior of the Canadian Arctic in search of the Northwest passage, rather than heading straight north, so Franklin in the 1840s did not go as far north as Ross in 1818.

There is archaeological evidence of earlier groups who lived further north than the Inuit, the Dorset culture, but that was during the medieval warm period and hence the climate is believed to have been more favourable.

Sources:

Narrative of an Attempt to Reach the North Pole: In Boats Fitted for the Purpose, and Attached to His Majesty's Ship Hecla, in the Year MDCCCXXVII, Under the Command of Captain William Edward Parry - William Edward Parry

Fury Beach: The Four-Year Odyssey of Captain John Ross and the Victory - Ray Edinger

Martin Frobisher: Elizabethan Privateer - James McDermott

Narrative of a voyage to the Polar Sea during 1875–76 in H.M. ships 'Alert' and 'Discover' - George Nares

Arctic researches, and life among the Esquimaux : being the narrative of an expedition in search of Sir John Franklin, in the years 1860, 1861, and 1862 - Francis Charles Hall