Was there a big difference between Labour and Conservative governments, for example... or not really? To what extent were British responses to independence demands based on the political ideology of the Prime Minister, or were they more a civil servant thing? How much pressure from the British people towards the government was there to decolonise or not?
This is a very complex and loaded question, because the decolonization movements span 3 continents (Africa, Asia, Americas). Also, the Conservatives were the governing party from 1951-63, during which most of Africa was decolonized.
There definitely was a big difference between Labour and Conservative attitudes towards decolonization, with Churchill getting a special mention for being staunchly against it. Labour had started the process of decolonization in the late 1940s, with the first wave of decolonization in Asia. However, Churchill and the Conservatives put a stop to that after coming back to power in the 1950s.
In his first term, Churchill was opposed to the parts of the Atlantic Charter pertaining to self-determination. Churchill rejected universal applicability of self-determination for subject nations. He even claimed that these terms pertained only to the territories of the German empire.
In his second term, the British refused to budge and instead doubled down on the wars it was fighting - e.g. in Malaya as well as Kenya (Mau Mau).
However, after Churchill's retirement (1955) and the Suez Crisis (1956), we start to see a paradigm shift in the British attitude towards decolonization. Ghana's independence (1957) would have a domino effect across the entire continent. This, incidentally, coincided with the government of Harold Macmillan.
Labour and Conservative attitudes would diverge significantly after this point, with Labour definitely pro-self-determination and some Conservatives still trying to justify holding on to their colonial possessions. However, after (Conservative PM) Harold MacMillan's [Wind of Change](https://en.wikipedia.org/wiki/Wind_of_Change_(speech)) speech, there really was no turning back for the British empire in Africa.
However, there was a faction of the Conservative Party which continued to oppose this push of decolonization, with the aforementioned speech directly leading to the formation of the Conservative Monday Club. The CMC even accused Macmillan of pushing the Conservative Party to the left.
Once Wilson (Labour) came into power in 1964, he continued the decolonization and extended it to most British colonies in Africa and Asia. By 1968, the process of decolonization of the British Empire was largely complete.
Source: Attitudes of British Conservatives towards Decolonization in Africa, Dan Horowitz, African Affairs Vol. 69, No. 274 (Jan., 1970), pp. 9-26
Party Politics and Decolonization: The Conservative Party and British Colonial Policy in Tropical Africa 1951-1964, Philip Murphy