'Conservative with a small 'c'' is a term used in Britain to identify people who are naturally conservative, but who are not necessarily supporters of the Conservative Party. It derives from the fact that the Conservatives, who were once normally called 'Tories', share their name with a state of mind that might not necessarily accord with their current political philosophy{1}. In the last couple of elections the Conservative Party, particularly on Europe, seemed less conservative than the Labour Party, and for a short period in the 1980s 'conservatives with a small c' might well have voted for the SDP rather than for a Conservative Party that seemed to be on a political crusade.
Although very few people in Britain - particularly in Scotland - would admit to being Conservative, most people are essentially conservative with a small c. In modern history the country has not had an extremist government, of any political persuasion{2}; neither the Communist Party of Great Britain nor the National Front, to pick two examples, are of any electoral significance. Of the major parties, only Labour in the very early 1980s came close to extremism, although it subsequently moved back from the brink. Nonetheless it is fascinating to speculate on what might have happened if Tony Benn had become the party's leader, and the novel and television series 'A Very British Coup' examined how the Establishment might have fought back against a democratically-elected extremist government.
As with the ironclad battleship, the corporation, and Monty Python, the term has spread to the former colonies across the Atlantic, even though the political parties over there have different names (and are both essentially conservative with a small c).
Some people might use a small word beginning with a small c to describe the Conservative Party, but that is not for me to say.
{1} A similar problem arises with the word 'liberal', of which much has been written elsewhere. Until a few decades ago Britain had a Liberal Party, and although its name has gone through a number of confusing permutations in the last few years they are still generally referred to as 'The Liberals' (or 'The Lib-Dems'). The party iself has historically been less left-wing than the more electable Labour Party, although the ideological collapse brought on by New Labour has seen the Liberals lose much of their identity.
{2} One wonders what could bring about an elected extremist government in the UK, although this is outside the scope of this writeup. Each successive political generation tends to be a backlash against the last, and the political pole bounces back and forth within a narrow range, never breaking out in either direction. The three most likely scenarios woult be a horrendous world situation, utter economic collapse, or an excessive reaction against an existing and unpopular party. Or, alternatively, if a party was to remain in power too long, it might become corrupt, entrenched and complacent.