Why nightmare? OTL gave part of world to commies.Japan winning certainly would be better.
And,i am taking about attacking dutch during Battle of England - Churchill could do nothing then.
Attacking soviets later - they could win and take Siberia,then made separate peace with soviets.
Sralin offered peace to germans at least till 1942 - but greedy idiots ALWAYS wonted moar.
Smart Japan here would say :OK,we take half of Siberia" ,and never join war on german side.
Britain has to act, albeit with very little simply because:
a) the Dutch government in exile, who's main base is the DEI is an ally. It would undermine our entire stance in terms of fighting to defend small nations to abandon an ally and also undermine the will of other allies to continue the fight.
b) Given that there are important other allies [i.e. ANZ] and many important colonial possessions Malaya, Burma, India etc that would be extremely vulnerable to further Japanese expansion once they have the DEI and imperial Japan has already shown itself as an extremely vicious and expansionist racist regime only an utter idiot would not expect that Japan would attack British interests as soon as their ready.
c) Even in the unlikely event of Japan occupying the DEI and stopping there even the threat of it would tie up a lot of allied resources. Think Australia would be willing to send virtually its entire professional army to the ME region with Japan on its doorstep? Or that India wouldn't have to deploy its own forces eastwards?
Because your talking about a situation where two of the most brutal and murderous regimes in the 20thC are even more successful and destructive. In worse case scenario Britain is defeated and the Nazis and Imperial Japan dominate virtually all of Eurasia for several decades at least. Death toll would be ~100M+ at least before they finally collapse. N America might survive as an island of civilization but then your could see too many fools thinking they need to go the dictator route as well to 'defend themselves'. Of course in such a scenario 'race enemies' such as the Slavs and alternative power centres, such as the churches are going to be among those who get hammered.
Slightly less bad would be Britain forced to make peace with Germany and able after a long fight to defeat Japan but even in the east the death toll is going to be appalling there and still going to be even worse in Europe. You might still get communism in China in this case as an exhausted Britain isn't going to be either able or willing to try and maintain some sort of order there.
Best case is that the US realise they have to get directly involved in which case Britain is already weakened and neither US or Japan are really ready for this war. Japan is more prepared but the US have much, much greater resources although in this scenario their will to fight could be markedly less. Coupled with the fact that a US dow on Japan would mean the loss of the Philippines along with Wake and Guam and possibly a bigger naval disaster than Pearl Harbour its likely that they could insist on only really fighting the Japanese or making the Germans a lesser target. Even if they do get serious about Hitler the power most likely to be helped by this would be your beloved Soviet Union as Hitler might not have time and an excuse to attack it.