So, I assume this one has been discussed before, since it's an old perennial, but a search didn't help much, so I figured I'd throw it out again.
I've been reading Roy Jenkins's biography of Churchill, and Jenkins buys pretty strongly to the argument that Halifax would have made peace with Hitler in June 1940 if he'd been PM. I'm less certain.
It certainly is true that, in OTL, Halifax was interested in approaching Hitler through intermediaries to find out what peace terms he might offer. This was very strongly opposed by Churchill as well as by Attlee and Greenwood, the Labour members of the War Cabinet. Chamberlain was more equivocal, but eventually (in Jenkins's account, at any rate) came to that position as well.
So, what would happen if Halifax, instead of Churchill, had been prime minister? I think there are two questions that have to be answered a particular way for this to result in a British peace with Germany. Firstly, would Halifax have actually carried out his idea if he'd been prime minister; secondly, if he did, would this have actually resorted to a separate peace?
So, on the first question, it will depend somewhat on the composition of the War Cabinet. It's quite possible we'd have exactly the same membership as OTL, except with Halifax as prime minister and Churchill as Foreign Secretary and Leader of the Commons (or Chancellor of the Exchequer and Leader of the Commons, I suppose, but I doubt Churchill would have wanted the Treasury). I imagine it's also possible to have a slightly larger War Cabinet of six or seven (the Chamberlain war cabinet of eight or nine was seen as too large, so we're certainly no larger than that). That might very well involve another Chamberlain loyalist, like John Simon or Kingsley Wood or Samuel Hoare, remaining in the war cabinet, which would of course make it more likely that Halifax could carry the war cabinet with him. Let's assume, though, for the sake of argument, that it's basically the same War Cabinet, but with Halifax and Churchill switching positions, since that's simplest.
But it seems to me that Halifax would have a very tough time carrying the war cabinet along with him. Churchill and the Labour ministers would almost certainly resign if Halifax insisted on it, which would put the government back in exactly the same place it had been on May 10, except with Churchill now an enemy as well. That's a dangerous place to be, an arguably, a completely insupportable one. The Halifax government, after all, had been formed largely to bring Labour into the government. To have Labour, along with Churchill, by far the most popular Tory at the moment, quit the government within a few weeks of its formation would be an utter disaster. Halifax would essentially have to admit to leading a narrow, quite unpopular, entirely Conservative government, staffed almost entirely by people who would be blamed for the military disasters which are still going on at the end of May. And he'd be doing so from the House of Lords, with his Commons Leader just having resigned from the War Cabinet.
So there's some reason to doubt that Halifax would possibly be able to push through his policy at all. He might technically have the numbers, given the large Conservative majority in the Commons, but he'd be risking mass defections.
But let's say Halifax manages to convince Churchill and Attlee that learning what Hitler's terms would be was not akin to surrender, and should at least be hazarded to see what terms might be on offer. (This is what he was actually saying at the time in OTL, so presumably he'd be doing this in the ATL, as well). So they don't resign, and the attempted parley goes through. What is Hitler going to offer? Trying to look into this, it doesn't seem altogether clear. But it's hard to imagine him offering anything that even Halifax would find tolerable, much less the more hawkish ministers the PM is desperately trying to keep on board. Hitler is not a man noted for the generosity of his proposed armistice or peace terms.
So it seems to me that the most likely outcome is that Halifax is basically persuaded to abandon the attempt, or alternately, that the attempt is made, but that Hitler's first offer is so terrible that Halifax has no real choice but to reject it and continue the fight.
What would change this would be a disaster at Dunkirk, but that very well might have done so even with Churchill in power.