DMcG
it was between one part of the Establishment (Churchill) and another part (Halifax and co), with 'the people' hardly getting a look in as usual.
Wiki,
"Halifax's implication was that if Churchill would not accept an attempt at peace, he would be forced to resign. If Halifax had resigned, most likely Chamberlain would follow, and then, Churchill would have faced a parliamentary revolt from the Conservative Party in the House of Commons, which could have led to his resignation as Prime Minister and the re-appointment of Chamberlain or possibly to the appointment of Halifax.
Faced with the threat of Halifax's resignation Churchill retreated from his hawkish position:"

That strongly suggests that most of the Establishment was with Halifax.

Churchill wrote, " I was sure that every Minister was ready to be killed quite soon, and have all his family and possessions destroyed, rather than give in. In this they represented the House of Commons and almost all the people. It fell to me in these coming days and months to express their sentiments on suitable occasions. This I was able to do because they were mine also. There was a white glow, overpowering, sublime, which ran though our Island from end to end."
https://en.m.wikipedia.org/wiki/May_1940_War_Cabinet_crisis

There were no anti-war public demonstrations or any indications that the people who would have to do the fighting and suffer the economic consequences did not support the war.
Few would now argue that they were wrong.