Some possibilities to put Britain in the mood to negotiate in summer of 1940:
(1) In the prewar period, the British heavy bomber advocates are successful in convincing the government that bombers will always get through and deliver the "knockout blow" that governments at the time feared. As a result, the British build up a bigger force of heavy bombers as a deterrent. Big fleets of four-engine bombers were expensive, and given limited financial resources, that build-up comes at the cost of having fewer fighters, not having the radar chain completely in place in the summer of 1940, getting a later start building up their army from the very low interwar levels, and some reductions in the naval build-up, which slows the expansion of their shipyard capabilities. When war comes, the big planes prove much less capable than expected for at least the first couple of years.
(2) France and Britain look at US neutrality laws in 1937-38 and conclude that they cannot depend on the US as a source for planes for expanding their air forces. As a result, both countries put their efforts into expanding domestic production, rather than placing the large orders they historically placed in the US. They also don't invest money into helping US aircraft manufacturers expand their facilities. As a result, the US aircraft industry does not expand anywhere near as fast in 1938-1940, and while the Brits have more production capacity for big bombers, they can't depend on the US for a flood of new planes in 1941 if they survive the summer of 1940.
(3) Weather permits the German attack in the West to happen in mid-March, preempting the Norway invasion and putting the Fall of France into a period where Chamberlain was still Prime Minister. The attack on France works as it did historically, with the exception that the Brits get a smaller percentage of their already smaller army out of Dunkirk, or wherever they end up trapped. Norway remains neutral, and the large Norwegian Merchant Marine does not become part of the effort to feed and supply Britain.
(4) The Germans capture French code-breaking facilities/personnel or documents that tell them that Enigma is compromised as part of the fall of France. That dries up an intelligence source that historically told the Brits that the Germans weren't ready to launch an invasion right after the Battle of France.
(5) Italy (a) Does not declare war until the bulk of its merchant marine is safely in Italian ports, avoiding giving the allies a gift of a third of the Italian merchant marine. (b) Concentrates all of its admittedly meager military assets against the Brits instead of spending most of the summer of 1940 building up for an attack on Yugoslavia that Hitler ultimately vetoed, (c) Sends the planes they sent to the Battle of Britain to North Africa instead, (d) Gets the M13/40 into production about 6 to 8 months earlier than they did historically, which means they have a few hundred available in Libya by the summer of 1940.
(6) The original Shah of Iran senses British weakness and nationalizes the British oil industry in Iran.
That's a lot of bad rolls of the dice for the Brits, but none of them are impossible and given the difficulty of getting to the Brits negotiating, multiple PODs are necessary to make this even remotely feasible.
Put all of that together, and you end up with the Germans sitting in the low-countries in mid-April 1940, with six months to potentially invade before weather in the channel made it non-feasible, as opposed to three months and change historically. Chamberlain is still in charge. He is blinded by the loss of Ultra and doesn't know how quickly the Germans are building up for an invasion. He has an incomplete radar chain and far fewer fighters (and fighter pilots) than the Brits did historically. British production of new fighters is lower than it was historically because it takes time to switch from bomber production to fighters.
Less fighter opposition means that the Germans can bring in dive bombers without unacceptable losses, which means they can bomb much more accurately. Poorer British radar coverage means that British planes can't intercept as accurately and are more apt to get caught on the ground. The German navy is intact, rather than nearly destroyed in the Norway invasion. Germany has used their airborne capability for the first time in Holland, rather than tipping their hand on its capabilities in Norway. As a result, their airborne forces suffer far fewer casualties in Holland than they did historically, and maybe their attempt to capture the Dutch government works.
Even with all of that, an actual invasion of Britain is unlikely to succeed. If (a) The Brits keep their nerve, and (b) The United States is politically able to support them financially, the Brits can still hold out. The Italians logistically couldn't do much actual harm to the Brits in North Africa. The best they could do would be to deny the Brits morale-building victories.
Would the Brits keep their nerve with Chamberlain in charge and a long summer of good invasion weather ahead, with the RAF losing control of the skies over Britain and with far less army to build on? I don't know.
Would the US be politically able to financially support Britain when it ran out of hard currency? Historically, Britain was essentially out of hard currency when Lend Lease passed in March 1941. I think they had to borrow money from either the Dutch or Norwegian government in exile to last that long. If they were fighting alone another two months plus change that could put them out of money perilously close to the US presidential election in November 1940. If the Iranians can make nationalization stick, even for a few months, that could run the Brits out of hard currency before the election. Not sure if Roosevelt could or would take the risk of proposing Lend-Lease before the election.
), when in fact they were able to keep the evacuation going long enough to retrieve about ten times that number.