After the death of Charlemagne, let's say that Louis the Pious does conquer Denmark in a Saxon Wars-like campaign. Although many of the inhabitants would remain crypto-pagan in beliefs for decades or centuries, I think we could presume that by the 880s-890s AD or so, Denmark would be Christianized in a political/diplomatic sense.
That works, as long as Louis' sons don't cause trouble for him the way they did IOTL. A pretty big divergence on its own, but I can't see Denmark being defeated Saxon-like by 840 otherwise.
Let's also assume a relatively convergent partition of Francia, except that Denmark becomes an appanage of East Francia -- probably a "Duchy of Nordmannia"?
This depends on Louis' three sons: if they fight each other as in OTL (840-42 was a pretty big civil war), the Danes will likely be able to wrest some control of their lands back from Louis the German - whether this would simply be greater autonomy or full independence is hard to say without knowing how far Louis the Pious went in his efforts to convert the Danes (I expect him to be milder than his father, but not excessively so). Independence will likely result in a Christian kingdom but with significant rights for pagans (whoever leads the rebellion would be the one to decide how far these rights go), and a Danish identity will survive into the X.century. Autonomy may result in eventual independence or Jutland becoming a part of Germany.
The problem with Louis' sons is Lotharingia: the most valuable third of the Frankish division is also the hardest to defend. If the partition of 840 looks the way it did in OTL, Louis the German will be much more focussed on Lotharingia than on Denmark, because something like Aachen is much more valuable than Danish villages that don't exactly want to be ruled anyway.
However.. if we assume Louis the Pious divides the kingdom more to how it looked in the 870s, civil war can likely be avoided. That is:
West Francia having Neustria, Austrasia, Saxony and most of Lotharingia
East Francia having Denmark, Bavaria, eastern part of Lotharingia and the northern parts of Burgundy
Middle/South Francia having Aquitaine and Italy (and a reasonable amount of territory to connect the two)
That being the case (and assuming there's no civil war anyway for unrelated reasons), Louis the German can probably finish off the conquest of Denmark (less a war at this point, but the occasional crushing of rebellions). The Obotrites will likely still be an EF ally against the Slavs/Moravians/Magyars (OTL there's little mention of the alliance after Charlemagne's death, but considering their hostility towards the Danes it likely continues ITTL). I'm unable to say how strong Louis the German would need to be to counter the Slavs (most of my research is in pre-800 stuff), but for simplicity's sake the Obotrites can fill in for EF troops in Denmark if needed.
What ramifications would a mostly Christianized Danish realm have by 880-890 AD?
Most obviously, no Great Heathen Army. And generally, less Vikings overall due to the fact that the Danes aren't contributing men. The Swedes could still oust the native Russians (Chuds and the like), but butterflies have a much larger role in deciding this than anything else. Denmark likely becomes the frontier of 'civilisation' just as eastern Germany did (until the conversions of Poland and Hungary around 960-1000), and there's no real reason to expect Christianity to spread much faster east than OTL as a direct result of this (unless the Church simply becomes more militant overall, but that's another discussion altogether). Of course, by 1000 you're looking at a totally different world anyway.
By the 900s, is it likely for the Danes to be recruited as mercenaries on the eastern frontier of Germany, particularly with the rise of Magyar raids? Could the Franks re-occupy some of the Danube fortresses left by Great Moravia?
By 900, Denmark is just as much 'German' as somewhere like Bavaria. There's no reason to expect them to be treated differently at this point.
Re Great Moravia, in the 880s the Karling dynasty had a pretty crazy collapse - Louis the Stammerer, Louis the Younger and Carloman II all died quite young, and Charles the Fat inherited everything only to die himself not much later. In all likelihood, this event doesn't get repeated ITTL. If it doesn't, Francia will look very different by 890: possibly divided, possibly in a massive civil war, possibly reunited and stable under another Pippin III/Charlemagne character. Under some of these scenarios, a march to Hungary is possible, in others not so much. Which scenario arises won't depend on Denmark, but on whatever alternate kids become Charlemagne's great-grandsons and great-great-grandsons. ITTL, Danish people will be a part of whatever the later Franks do, because they're now 'Germans' not 'Danes', but nothing more can be said with certainty.
Al-Andalus is a wealthier area to conquer, but the king of East Francia may not have direct access to it, excepting a reunification of the Frankish realms.
The general trend of the Franks, going as far back as Clovis, was to only fight external enemies if Francia was totally united (see Charlemagne not invading Saxony until his brother died IOTL). If the King of East Francia is not also King of West Francia, he's going to be worrying about what WF is possibly going to do to him - not if he can go conquer something else. Sure, if the Slavs attack him he'll defend himself, but he's not going to march an army deep into Poland if it risks his cousin attacking him for the throne.
- BNC