What are some examples of rulers immediately incorporating conquered population's army into their own?

The question is extremely vague because the main concepts are too generic (conquering armies, conquered populations; and that covers the entire time span of war history up to the present); different moments involve very different scenarios: there are great differences between the ways in which such a phenomenon could have occurred is a tribal union, in the Chinese empire, the Roman empire, the medieval monarchies and the modern national armies. But the general fact of using the armies or soldiers of a formerly defeated foe as part of future actions is the rule rather than the exception. The ways in which this was done varied of course, depending on very different material, political and cultural factors.

Migratory conquerors from the Eurasian steppe (from Sarmatians and Scythians to Huns, Khazars, Goths, Cumans, Turks and Mongols) were not only comprising various tribes dominated by some that had been the victors over the rest, but the very power of such famous invaders - one that sometimes seemed to vanish shortly after a major defeat - was based on the multiplicative force of secondary tribes whose names have either been forgotten or have been famous at a previous or at a future moment. For example, Goths have been defeated and chased away by Huns, but the Hunic army was a mixture of various migratory peoples, including Goths. When the Huns have been defeated, Goths played a major part, but Goths were in fact fighting on both sides. When Huns were defeated they seem to disapear from chronicles, but most probably their armies (which were not purely "Hunic" to start with) were reintegrated within subsequent ones. (see Wikipedia on Steppe empire dynamics and ethnogenesis.)

Ancient Egypt used soldiers of conquered and client states, beside mercenaries (Wikipedia on that here and here).

In fact mercenaries and slave-soldiers are by definition re-employable military forces.

Ancient Middle East empires, from Sumerians and Assyrians to Persians provide various scenarios of interaction with defeated armies (from extermination to integration). A typical case is that of slave warriors which were present in the area from the ancient Persians to the medieval and modern era (see Ghaznavids and Mamluks). Assyrians typically used terror and deportation, but also armies of mixed origins to strengthen their grip (also here). Persian empire had a multi-national military structure that was later inherited by Alexander the Great.

In the Roman empire (beside the well-known use of foederati) an example is that of the auxiliary forces of various origins. Sometimes these forces came from fully conquered peoples; (an extreme case is that of Dacian forces - Dacia was conquered in 106 AD - used to build and guard Hadrian's wall in northern England: see Cohors I Aelia Dacorum); sometimes they came from peoples defeated but not conquered; during the entire history of the Roman Empire (including its Byzantine form) on innumerable occasions treatises/pacts with (sometimes) defeated "Barbarians" (from BC-era Germanic, Gaulish and Iberian tribes to medieval Bulgarians and Turks) stipulated the provision of barbarian troupes (especially cavalry) to fight on the Roman side.

As a general rule empires tend to integrate conquered populations within their military structures. This is true from Assirians to Romans and from British Raj to the Soviet Union. All colonial powers used "native" military.

Mongol armies were multi-ethnic from the beginning, and further conquests led to the integration of Chinese, Turkic, Persian, Armenian, Georgian, Slavic and other forces. (When Mongols attacked Hungary they did under the pretext that Cumans had taken refuge there: as if Cumans belonged to them, rather than to Hungary, in the sense that the Cuman armies should have become part of their army. - That argument seems related to the aforementioned "steppe empire dynamics and ethnogenesis" linked above, as it was used multiple times in history, for example by the Göktürks claiming rights over the Avars in their negotiations with the Byzantines in 565 AD.)

Arab and other Islamic conquests were helped by accelerated conversions to Islam, in turn encouraged by the prospect of renewed military carriers. This is another example of rapid integration of previously defeated military forces, although in many cases these fighters were slave warriors or mercenaries.

Examples are endless.

Vassalage in medieval Europe is another special case of "using" or "integrating" previous enemies into own armies. (Think of armies of the duchies of Burgundy, Normandy, Brittany and so forth after their direct integration within the French monarchy.)

China has a long history where this logic can be seen at work too, and Japan is another case. Just like in the cases of Middle East empires or of Egypt, their history is too long to be discussed here in detail.

From imperial China to 19th century Germany and Italy universal conscription (or conscription of commoners) in territories annexed by military means can be put forth as another example.

Napoleonic armies that attacked Russia were largely (more than 1/3) formed of Polish, Italian and German soldiers. Later, African and Arab soldiers fought for France in both World Wars, just like the Indian and other forces did for Great Britain.

Nazi Germany used foreign conscripts of French, Dutch, Ukrainian, Russian and many other origins (many of which had been part of defeated armies). Defeated Germans were later mobilized against USSR in the West Germany and against NATO in East Germany: theoretically these were independent national armies but they were created based on previous defeat and occupation. Like the Russian army before 1917, the Soviet Red Army included from its beginning Baltic, Ukrainian and other forces from nations that were in fact under Russian domination. (The situation is more complex of course.)

Even in very recent times the United States have occupied countries and mobilized local forces under their direct control. Afghan and Iraqi armies were reorganized in order to fight along (and also in place of) American forces. (Again this is another very particular variation on the theme.)

See HERE a short video about former Waffen-SS Baltic soldiers incorporated by the US army. Also - Wikipedia.