I assume that by Medieval you mean Western or Central European Medieval, say around the 10th to the 12th century. People often forget that history happened worldwide and that the Middle Ages lasted for a thousand years... I also assume that by "kingdom" you mean a generic monarchical medieval state; there were not that many actual kingdoms in medieval Europe -- most states had lesser titles, such as principality (ruled by a prince or some native title, e.g., voyvode), grand duchy (ruled by a grand duke), duchy, march (ruled by a marquess or some native title, e.g., margrave), county etc.
Theoretical considerations
So, a European Medieval state suddenly finds its population and territory ("kingdom A and B unify as one kingdom AB") doubled. Big deal. Unless you are speaking of one of the very few large countries (initially only France, then England, then Spain), medieval states were perpetually short of inhabitants: low workforce, low tax base, tiny army... This runs against our modern perception: we live in a world with a surfeit of humans; but in the Middle Ages Europe had very few people, even without bringing the Plague into discussion. A locality with 15000 people was a very large city; the typical town had maybe one thousand, usually fewer. There was ample uncultivated and unclaimed land -- in most places, if a group of people banded together and cleaned off the forest to start agriculture the law and custom gave them title over the land.
For example, let's say that the Barony of Parchim united with the County of Schwerin, resulting in the Landgraviate of Parchim-Schwerin: without using Google, do you even have the faintest idea where was the Barony of Parchim? Yet it existed, and it was a member state of the Holy Roman Empire of the German Nation; and yes, it had a border with the County of Schwerin.
But what if you are speaking of two largish countries? Say that Denmark and Norway and Sweden got united. Guess what, they did: it was the Union of Kalmar, which lasted from 1397 to 1523. Nothing much came out of it.
You see, in the European Middle-Ages states did very little for their inhabitants. Essentially, they did exactly three things: they provided justice (including record-keeping, the most important thing they did), they provided defense (as well as they could), and they imposed social peace (sort of, mostly by hanging thiefs and murderers provided somebody caught them and brought them to justice). Nothing more. No social security, no pensions, very very little in terms of infrastructure (mainly the occasional bridge), no education, no health care; what little education and social security were available, not much, were provided by the Church. Remember that medieval states were perpetually starved of workforce and tax base.
Moreover, there was no concept of "nation" in the Middle Ages, or, to put it in a better way, the word "natio" (Latin, from nascere, nat-, to give birth) simply meant "people who speak sort of like the same language" and had no political dimension. It was perfectly reasonable for a state to have inhabitants speaking different languages; it was actually expected that peasants from different provinces wouldn't be able to communicate: they were peasants after all, and did not know Latin, which was the only actually useful human language; administration was done in Latin, justice was done in Latin (except in England, were justice was done in an obsolete form of French for some reason), religious services were done in Latin. Nobody cared what language the peasants spoke, if the noises they made could even be called human speech. And linguistic identities were very fluid; people were not sentimentally attached to languages, and were perfectly willing to learn the dominant language of the place.
All right, but was would happen in practice?
In practice what would happen is that a handful of noblemen from the newly incorporate land would present themselves at the court of the new ruler, and perform homage, declaring themselves vassals of the new king. They would then have a talk with the treasurer of the new king, who would incorporate the revenue of their lands in the tax base, and possibly with the top generals, who would take their levies into account for the host. Everybody would be speaking Latin, of course, or some sort of German, or French, or Toscan.
Some time later, maybe a few months, maybe one year, some noblewomen from the newly incorporated land would appear at court. They would likely speak some sort of German, or French, or Toscan, or even, shudder, the local language (women are feeble-minded, you see, and few of them know Latin).
And that's all. Ordinary people had very little interaction with the state, and couldn't care less whether they were ruled by the Freiherr von Parchim or by the Graf von Schwerin. Life, contracts, marriages, baptisms, trade, would continue exactly as before. Maybe the new united state would have a little more military power, maybe it will have a little better financial position. Maybe. The medieval world is in a large part a mosaic made up of many many tiny little pieces, which have little interaction between them. It is possible that the new united state will develop faster, but it is not certain, and not even likely.
