I suspect that had Britain joined the Axis either through invasion or subversion, De Valera would have wasted little time throwing off neutrality and joining possibly with Ribbentrop offering Dublin a free hand in Ulster as an inducement.
Obviously, an overt German move into Ireland would have been viewed negatively in Washington so if De Valera chose to remain neutral, along with Sweden, Spain and Switzerland, it's probable Hitler would have respected that in the short term.
IF Ireland is under Axis control directly or indirectly, what are the options ? The British occupied the Faeroes after the fall of Denmark but if Britain itself is overrun, it's hard to see a British presence in Torshavn surviving long. America might ensure Iceland and Greenland stay out of Axis control and Portuguese neutrality (if it remains) might mean the Americans have use of the Azores.
The other question is whether the Empire is still fighting or whether all British military activity has ended. It's entirely possible that even if the homeland is lost, Canada, Australia and New Zealand and others (though not perhaps South Africa) might try and re-group using India or the Middle East as staging posts ? Would Imperial Forces (including any evacuated from Britain before it fell) resist an Italian incursion into Egypt ? Could Malta be defended or the attempts to defend Greece be made if Britain itself has fallen ? These impact directly on Barbarossa's timing and whether there is a North African theatre.
Doubtless Hitler will attack the USSR in 1941 - will Japan attack Pearl Harbor at the end of the year ? If they do, the US and USSR would find themselves as allies or co-belligerents along with the rump of the European empires. Can the Australians, South Africans, Indians, New Zealanders and remnants of the British stop Rommel ? If they can, would logistics enable Eisenhower to land at the other side of North Africa and push the Axis back to Tunisia and Libya ?
I'm stretching it now a lot but if the American Alliance seeks to prioritise the liberation of Europe, it might land forces in Italy and southern France in 1943-44 and seek to push north as the Russians seek to push west. As the Americans approach the Channel Coast from the south, would the German occupying force in Britain be faced by a large-scale Resistance uprising (the British Resistance has both Communist (in the cities) and Royalist (in the country) elements) ? Hitler would order his Army commander in London to stand fast but with American air supremacy and the Channel ports in American hands, the likely option is for the Germans in Britain to be isolated through 1945-46.
IF the A-Bombs are dropped on Berlin or elsewhere, the occupation collapses and the Germans surrender to the Resistance or the Americans (akin to what happened in Norway in OTL). I struggle to see a German-occupied Britain surviving once Berlin falls and most of Germany is occupied. It's possible with the American and British navies in command of the Channel a contingent of American or (more likely) Empire troops would be ferried across to Dover and Folkestone and march on London largely unopposed. The German troops would be quickly deported or interned and the collaborators punished. Would a victorious Stalin seek to incite a Communist insurrection in Britain or would he recognise it as being beyond his means and accept Britain would fall into the American orbit in the post-war division of Europe.