I have only read about Sweyn Forkbeard and Cnut the Great, so my answers will be based on my knowledge on them.

First of all it is important to remember that the Nordic invasions of England actually weren't composed of a lot of people. While the numbers to us are unknown, my knowledge of the estimate of the earlier Great Heathen Army of the 9th century *at most* numbered at a couple of thousand warriors. The same is the case with Sweyn Forkbeards invasion army. The numbers are not known, but they're estimated to also only be around the same size. Compare this to the then population of Denmark which was around 500,000 and it doesn't seem like it would have been difficult to find manpower for the invasions. The Chronicle of Gesta Cnutonis regis tells of Sweyn gathering his army for the invasion by laying forth his plans for his "chief men" and soldiers and that they approved of it. Sweyn had probably earned a good standing among his men due to having lead a few successful raids before the conquest.

Now where did Sweyn get his soldiers from? We aren't sure. Many or most were probably Danish, but the process of recruitment is unknown to us. It also seems to us based on later sources that his army was probably multicultural - that is made up of a lot of foreign warriors. This might suggest that there was some kind of inter-kingdom mercenaries who flocked to successful commanders in hope of loot, but that is only speculation. The soldiers also probably had some kind of relation to Sweyns "chief men", but its nature is unknown to us. They most likely commanded some of the forces as well and obviously had a say in how the realm and the army should be governed and used.

Sweyn ended up dying just one year after his conquest of England so we don't really know much of how he governed his realms. His son Cnut seems to have had the same army and navy as his father - as he was elected by them according to the Anglo-Saxon Chronicle and that he kept the loyalty of the army by paying them a lot of money which alludes to them being mercenaries paid for by the money both Swyn and Cnut had gained from their raiding expeditions in the past.

How did Cnut govern his large realm? I'm not very well read on this topic to be honest, so I'll only talk about Denmark. I'm not british, but I expect the source material for that area to be more comprehensive so I hope someone will come by with that perspective. Cnut became king of Denmark after his brother's death in 1019, but he mainly stayed in England. This doesn't mean he held power in Denmark. Mints were expanded during his reign to more cities and minted coins with his name and according to Saxo he made some of the first military laws - however it is possible Saxo simply claimed this to give the laws of Valdemar and Cnut IV a century later more legitimacy. Militarily he was also quite able to protect his Danish realm. From what I know Denmark wasn't under any threat at the time from the Holy Roman Empire, which I believe was busy with Italy and the Easter Roman Empire, but it was attacked by a joint Norwegian and Swedish attack in the mid-1020's. Cnut didn't have a standing army and for a time it seems that the Danes must have protected themselves until he could amass a navy which ended up beating the Norwegians and Swedes back after which Cnut also was able to conquer Norway.

In conclusion. We don't really know how these armies were organized and recruited, but most likely through alliances with certain "chief men" and recruitment through payment - either in the form of promised loot and/or through direct payments. These armies were possible to muster most likely because Sweyn and Cnut had gained a reputation for being successful raiding commanders and because they had a lot of money from all of their successful raids. But in the end we havde little idea exactly how Cnut was able to govern Denmark.