This guide covers one possible reliable way to get this achievement in about 1h at a good gamespeed. Thanks to historical focuses, the realtive easy difficulty of the achievement and really no random factors at all, it should be possible feel pretty easy.
If there are complaints, difficulties or comment, i appreciate any feedback!

Don't forget to turn off all mods and enable ironman mode.
Start the game as United Kingdom with Historical AI Focuses activated.

• Select the focus "A Change in Course"
  
• Research industry (Tools and Construction), land doctrine (superior firepower) and engineering
  
• Use your military factories primarely on Infantry Equipment and other needs for the infantry
  
• Group your fleet to make them effective at escorting divisions
  
• Group the airforce to restation them easily in the upcoming small scale wars
  
• Delete garisson templates in favor of building more proper infantry divisions
  
• Station the remaining armies at the canadian, the irish and the south afircan borders to your states
  
• Until "The Kings Party" focus becomes available, go for "Encourage Colonial Elites" and wait until Edwards takeover before taking any more focuses
  
• Stick to building civilian foctories at least until mid 1937

It is not possible to advance the needed political focses until Edward becomes King

• Some months into the game the "Edward VIII Abdiction Crisis" event will show up
  
• Select the "Insist on a royal marriage" option
  
• From here on there will be positive and negative events influencing party popularity and political power, in order to advance always "accept the result" and avoid options resulting in Edwards abdication
  
• The dominions will soon become independent, so don't station any division inside them
  
• As for the research focus on more industrial and infantry technologies, always research on the land doctrine
  
• Safe your political power for the coup d'etat and the following events

In order to have a stating ground and equipment for the invasion of the US it is important to conquer some territory and retake some dominions quickly

• Select the focus "God save the King" and go for "Appeal to Imperial Loyalists" next
  
• Upgrade your economy to "Partial Mobilization"
  
• Keep on upgrading or expanding your forces at the borders described in "Getting started"
  
• Keep at least 200 Political Power
  
• When "Appeal to Imperial Loyalists" finsihes, stage uprisings in Canada and South Africa immediately and go for the "Consolidate the British Isles" focus
  
• Once it finished annex Ireland fast and redistribute the divisions there to South Africa and Canada
  
• Technicaly it is not needed to capitulate South Africa, but it levels your armies, while newfoundland doesn't have the needed supply for your whole army
  
• Go for "Bring the Dominions back into the fold" in the focustree
  
• After the uprisings and the focus finish, ally the uprisings and help out the loyalists by attacking from a new front
  
• Enact better conscription laws and hire military advisors for division speed and infantry stats
  
• Since the Dominions don't have many or good divisions this should be fast and easy
  
• With Canada capitulating there will be a small peace conference which will make it (and South Africa) your puppet again, allowing you to attack the US

The Empire Strikes Back achievement included for free

• Continue the buildup with the new factories and experience, especially by training some more infantry divisions to cover the whole US-canadian front
  
• Station your whole army and airforce in Canada to allow an fast advanced against the yet to be mobilized USA (Request all of the Dominions forces to fill frontlines)
  
• Rush down the focuses towards "Unite the Anglosphere" (should finish arround late 1938)
  
• The second the focus finishes, declare war and advance aggresively
  
• Use your fleet as convoy escort if reenforcements or supply routes are endangered
  
• Considering the US will be unprepared it is easy to overwhelm them and encircle some division on the way, focus the Eastcoast to gain most of the US industry
  
• The war should be easy if there are any problems with that i recommend watching into the achiement run video attached below

Normaly there shouldn't be any more major poweres involved.
If the US joined Australia or France it is easy to take them over by simply invading them by sea.
(In my video i made Australia join, which simply extends the time needed)

Once the peace conference starts, puppet the USA and handle potential other defeated nations however wanted.
With the nation under your controle there will be a new decision und "Political Actions" called "Install American Monarchy".
Enacting this one and releasing Scottland as a puppet will reward you with the achievement.

If there are any problems while follwing the guid, especially during the wars against former colonies, I recommend watching this achievement run video to clear up any questions: