So this is a pretty huge question and it would really help if you could narrow it down to particular regions, dates, and peoples. e.g. by "the British" do you mean imperial administrators? Or the populace at large? By "natives" do you just mean any colonial subjects? Or tribal natives? And where/when? The views of tribal natives in America in the 17th century will be different from those of tribal natives in Africa in the 19th.

As it is, I can answer a bit in regards to some of Britain's Arab possessions in the late 19th century, but am limited to the views of some of the major figures rather than the British populace at large.

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Lord Cromer, Evelyn Baring, who was British Consul General of Egypt and oversaw it's occupation and rule for about 25 years wrote a pretty detailed memoir/history entitled "Modern Egypt" wherein he shares his views.

His basic view was that he was on a civilizing mission, that these were a "semi-civilized" people. He was sympathetic to what he viewed as the mismanagement of Egypt under the Ottomans and the Pashas and claims to have tried to advance their civilization as he saw it but basically he falls in line with the classically "orientalist" tropes. He's actually used as a prominent example in Said's "Orientalism."

He's not far off from Rudyard Kipling's view that "East is East, and West is West, and never the twain shall meet." He quotes Kipling in the book so he's clearly familiar with his ideas. He viewed Europeans as having an inherently rational mind which was inaccessible to Egyptians and vice versa, although he thought they were probably more European than Indians.

Towards the other end of that spectrum, and a contemporary (and opponent) of Cromer was Wilfrid Scawen Blunt, who wrote a "Secret History of the Occupation of Egypt" in which he details his working with Urabi for the cause of the Egyptian people in the face of British imperialism. Where Cromer clearly took some idea of the white man's burden seriously Blunt wrote in a line of poetry "The white man's burden, Lord, is the burden of his cash" indicting imperialism as a profit driven enterprise rather than one of cultural/civilizational development.

You can find these kinds of contrasts in pretty much all levels of the British imperial service right up until 1950s and the end of empire. In the following generation for instance you have the contrast between Arnold Wilson, who tried to impose India-style administration on Iraq as a prelude to its adoption into the Empire and Percy Cox who, with Gertrude Bell, T.E. Lawrence and others (including to some extent St. John Philby who is a fascinating figure in regards to your question, he was one of the British advisers to ibn Saud and basically "went native" converted to Islam and was called Sheikh Abdullah), tried to organize Iraq for some form of independence.