The British are CONQUERED per the OP (already noted the low order of probability on this scenario). They did not make a deal, they did not arrange to retain anything. The British got punched in the mouth and rolled just like France, Norway, Belgium, the Netherlands, Poland IOTL and the ATL USSR. Game Over.
British troops are now utterly unsupported. No aircraft, no ammo, no rations, hell, no new boots and socks. Assuming the Government in Exile manages to retain control of the forces, which if far from a certainty (the Reich does, it should be noted, have an actual King of England available to install back onto the throne) there will have to be a series of decisions made regarding where they make their last stand. There is simply no way to retain everything. Trying to hold all means losing everything. Unquestionably that effort would be made in India.
Without India Egypt has no value, nor does Iraq. India already has some issues with independence efforts, the division & a half in Iraq, along with the obsolescent aircraft will be much needed in India, as would the forces in Egypt. Unlike the forces in Egypt the forces in Iraq can readily move to India (which at this point in time included Pakistan) using local shipping (some movement could even be made overland along the Iranian Persian Gulf coast). Iraq has absolutely no value to the British in this scenario. The RAF has exactly ONE bomber in Iraq with the range to reack Baku from Mosul, a Bristol Blenheim. There are also 14 Vickers Valiant single engine biplane bombers (max speed 143mph, combat radius 525 miles)
It is also rather important to keep in mind what the Reich did regard France's colonies that another member of the Axis wanted. The French were required to stand aside and allow the Japanese to take control of French Indochina (and Berlin's relationship with Tokyo was far less cordial that the partnership with Rome). Mussolini wanted to recreate the Roman empire around the Med, Hitler was not going to stand in his way, IOTL Hitler went to considerable lengths to support his Italian partner, in this scenario it is impossible to believe that the Reich would not support Rome's demands from the British. Even if the British force decide to screw their navels to the ground and defend Egypt (which would be strategically idiotic for any Government in Exile in this scenario) the Wehrmacht will be sent to support the Italians, same as IOTL, except this time there will be no British fleet to send to support Egypt, no daring efforts to support the brave defenders on Malta, no RAF or RN forces on the Rock to provide support. Rommel (or whoever Hitler dispatches to command the Africa Corps) will roll the British up like a rug.
Iran won't be occupied by the Soviets because the Soviets got their ass handed to them ATL. They lost the war, almost certainly during the in initial Barbarossa Zerg-rush. The Soviet Union is done like dinner in this scenario. It won't be occupied by the British because the British don't have the ability to shift forces like that and it is no longer critical in the overall scheme of things, the Americans won't occupy Iran because there is no reason to. USSR lost the war, no reason for Lend Lease, Iran doesn't matter in this scenario.
The memo you attached is discussing the Mark III (i.e. Fat Man) weapon. The U.S. never put the Mark I Uranium (i.e. Little Boy) weapon into large scale production, primarily because it was vastly more difficult to produce sufficient enriched uranium. Only five Mark I bomb assemblies (physics packages) were ever produced, only one was weaponized and it was expended at Hiroshima.
There is no way that one can simply overlay OTL onto this scenario. It is entirely possible that the POD butterflies away the Pacific War. The U.S. embargoes against Japan, especially oil, were only effective because the UK and Dutch Government in Exile went along with the U.S. That is no where near as certain ATL. While the U.S. military build-up will likely still occur (if the Congress lost its mind after the Fall of France, any defeat of the British will have them pushing every panic button they can find). It is very possible that the U.S. adopts a "Monroe Doctrine" on Steroids, where America's umbrella extends to Australia and New Zealand, but it is also possible that the decision is closer to "New World is ours, stay on you side of the Atlantic and we're cool".
I would respectfully disagree with a lot of this, but sadly lack of time is really and truly a bitch.
Trying to keep everything reasonably short, IMHO:
1. the British government-in-exile will almost certainly use the carrot of future independence and the stick of a possible Japanese takeover to keep the peace on the Indian home front.
2. Edward, assuming he somehow sneaks into Europe, will be soundly ignored by any and all British territories not under Nazi jackboots
3. with German panzers pushing from Dover to Inverness in the latter half of 1940, Mussolini will not, under any circumstance, accept the humiliation of asking Hitler for help
4. given the above, any occupation of French North Africa prior to Barbarossa is simply not feasible
5. fighting in Russia is likely to last until December '41 at the earliest, with or without Stalin, due to the distances involved if nothing else
6. while the above might persuade the Japanese to 'go north', their fuel supply will not be getting any better, as european governments in exile would be drawn to the US orbit more than ever, making, IMO, any avoidance of a Pacific theater highly unlikely
7. Given 3-5, any Axis attempt to occupy French North Africa is bound to come in AFTER Pearl Harbor, making the possibility of an Allied intervention in the area in help of the French (who absolutely hate the Italians and would not accept being taken over by them) quite likely
8. Since his food situation won't be getting any better with Britain occupied anyway, Franco will likely drag out joining the Axis (since doing so means almost all crucial trade that still keeps Spain afloat will be cut off), meaning the earliest Germany can deploy a substantial force to Gibraltar will be '42, still providing a decent window for an Allied push into Morocco
10. For reasons which are too long to mention (disdain for Jewish physics; lack of heavy water; uncertainty over what, if anything, they capture in the UK; competing priorities; competing agencies working on a Nazi A-Bomb; false understanding of the process involved etc etc etc), I find it almost as unlikely as the OP of this thread that the Reich will produce a fission-type Bomb sooner than the US will (or even significantly close), with or without this or that scientist escaping from Scotland
11. Since we're postulating a collapsed Soviet Union, I think it's fair to make the assumption that part of what allowed Hitler to pull the whole scheme off was not deploying the DAK in the first place (probably due to no pressing need & no invitation from Benny), meaning Egypt will still be British in early '42