The Mongols conquering Europe schtik is one that has been talked about quite often but the details and extent of any such scenario are never really covered in detail or to any plausible extent such as it is unlikely that (in short order) the Mongols could as a singlular force roll through Germany and into France and Spain etc etc. Though even still that doesn't mean that the Mongols could have been more involved in Europe proper, as the Golden Horde was mostly concerned with control over the Volga region, its Russia vassals to a small extent but mostly quarreling with the other Khanates to the east. I want to keep the Mongol's eyes on the West.
Though besides working from a POD with the possibility of either Ogedei living longer or at least the news of it taking longer I want to look at the Capture of Bela IV following the Battle of Mohi rather then his OTL escape to Frederick in Vienna. Now in OTL he fled westward and was more or less extorted by Emperor Frederick II to give up his money and land to Austria (which I think sets a precedent for Austrian intervention and rule in Hungary) and made numerous pleas to Frederick and the Pope to call a crusade but nothing really happened as the Mongols pummeled armies of Germans, French, and Poles and rampaged through the Balkans... Lets say that Bela IV is captured, the results for him could be either execution or vassalage which is what I am looking at. Granted still quite abit of plundering would go on but if the Mongols can maintain a prescence in the Pannonian Plain and absorb the elements of the fleeing Hungarians and Cumans then they have quite the springboard into Central Europe.
If allowed to continue, the Mongols would likely smash open Vienna and launch attacks into East Germany which could result in some interesting succession possibilities for the title of Holy Roman Emperor if Frederick is killed or captured. You see at this point in things, even without making attempts to incorporate the Hungarians and Cumans that had fled them (the Mongols had absorbed Cuman tribes into their army at this point OTL) the Mongols had momentum (especially considering the less then united opinions the Europeans ahd toward facing the Mongols) and any field army the Europeans would send at them would more then likely be destroyed. The only thing that would proove a sticklier to the Mongol advances would of course be the fortified settlements and castles which could take time but the Mongols could ignore them entierly and loot the countryside or bring up gunpowder siege weapons (depending on the target, they would most likely move to take Vienna for example because of its prominence). In OTL it was only after the Mongolians left that Bela IV decided to intergrate Hungary into Western Europe and go on a Fortification-spree that did see a defat of Mongol forces in OTL.
A full physical conquest of Europe though is unlikely, and probably something that Subutai and Batu would realize (though future gains in Europe are not out of the question, they would need to marshal captured assets like Russia, Poland, and in this scenario Hungary as they did with Chinese soldiers in the east). As I have mentioned what the Mongols could very well do for sometime is make raids into Central Europe and probably Northern Italy. Through these raids, their past victories, and their own well known inflated public image Batu would likely issue demands for tribute and vassalage which at this point I can see him making vassals out of the various Balkans states and the Hungarians and Poles and whoever claims the remants of lower Austria and Bohemia...and probably regular tribute/don't-attack-me bribes out of the German Princes and possibly Venice and Milan.
Such a scenario would herald in Turkic Migrations from the Volga Steppe, just as had been done with the Vandals and Goths by the Huns and so was done with such groups as the Cumans during their time. This would probably be focused in Hungary and the northern Balkans as well as Poland and Russia. Though of course Mongolian too but I do believe that the Mongolians themselves formed a elite unit within the much larger Turkic Speaking Army.
Of course one would have to address Ogedei dieing eventually, so either the Golden Horde takes a much more westward attitude or a seperate section of the Golden Horde is formed with one of Batu's successors looking eastward while the other looks westward and likely south to Constantinople, subjugate or ally with the Bulgars and take in the situation of the VERY precarious rules of the Crusading States of the Latin and Thessaloniki Empires...
Now with the Mongols apparently there to stay on the doorstep of Europe, what would reactions be in Western Europe?