No, you didn't - you avoided being invaded and subjugated, but you certainly didn't stop them, because they kept right on going. So far as Hitler was concerned, you were stuck on your little island, going nowhere, so he could turn to projects closer to his heart, such as the creation of Lebensraum for the thousand-year Reich. I know it's part of the British mythology of plucky little Britain standing alone, but it wasn't alone - it had the resources of an Empire behind it.
I know Churchill said in one of those wonderfully melodramatic speeches that, if Britain succumbed, even the USA would succumb, but I doubt it. The fact that the USA was on the far side of a vast ocean frustrated the life out of the Nazis. They would have loved to have struck at the US cities, but never managed (the Japanese fired some rounds at US west coast cities and facilities, and even dropped a few bombs from submarine-carried floatplanes, but they were never more than pinpricks).
The other thing was that the USA was not merely a rival country but a rival continent, with the means of production far exceeding that of all of Europe, which, once it got going, would produce Nazi Germany into the ground. In the aftermath of Pearl Harbor, Admiral Yamamoto said it best - I fear that we have wakened a sleeping giant and have filled him with a terrible resolve. Hitler very foolishly did exactly that, and paid the price.
I don’t believe it to be anything like that simple.
My logic being that if the UK had capitulated after say the invasion of France:
- Neither the RAF of Luftwaffe would have had the losses during the Battle of Britain. For ref, approx 1/3 or all Luftwaffe air crew were lost
- The Tizard mission wouldn’t have happened. That means no cavity magnetron, no jet engine, and no chaps from Birmingham convincing their equivalents in the US to build the A bomb
- you may have noticed how effective the German army was in assimilating troops from other countries. I could very much picture them doing that to the Royal Navy, RAF and army.
- when the Germans invade Russia, they now get to call on not only their own army, but also the British army. More importantly would be the extra air crew and aircraft from the RAF and those not lost in the BOB. Barbarrossa is successful with the additional resources available and Russia is taken under Nazi control
- the British empire hands over access to the oil in the Middle East and has the merchant navy to move it
- uranium for the German bomb can be sourced from ex British colonies
- The Royal Navy patrol the Atlantic and Indian Ocean to prevent shipments to and from the US
- the British don’t source armaments from the US. The requirement for the P51 for the UK never happens and packard aren’t given the designs for the Merlin engine
- British and German engineers collaborate on jet engines and the Germans have access to the alloys required to enable their jet engines to run for more than 25hours before melting
Take the above as reasonably likely scenarios, would the US even bother coming out of its slumber of non-interventionism? I wouldn’t have been surprised if the hard right in the US backed Hitler, just as some now consider Putin to be OK.
If the US did fight, the Nazis could choose where to land, would have a lot of combat experience vs not much, and might have waited until they had jets and the A Bomb. The background for the Man in the High Castle is not looking too unreasonable