Gregory's Server'Conservatism' with a small 'c' is one of the flavours of centrism - centre-right, as opposed to centre-left - but the basic proposition of both is: 'Things are not too bad, change is risky, let's just tinker a bit (centre-right) or a bit more (centre-left).
But of course actual parties that call themselves 'Conservative' vary in different times and places. In the UK the 'one nation' strand within the Conservative Party was dominant before Thatcher - this was genuinely centre-right - but with Thatcher the more extreme right took over, so it's now generally regarded simply as right-wing.
Before brexit, you could chart this movement via its affiliation in the European Parliament, moving away from the mainstream centre-right grouping and joining the far right.