First of all, the Scandinavians were pretty good at sailing the northern seas and had good trade connections within the region. The regions that share a language and are not isolated from each other usually don't diverge into different languages.
Second, when the split started occuring, the regions were held together by the Kalmar union, and as such unified under one monarch and one administration. Surely the monarchs would've seen the divergence and tried to hold the language in its unified form for simplicity of administration - so why didn't they?
Old Norse had already diverged into West and East Norse branches by the 10th century, and by the start of the Kalmar Union those had diverged further into Old Icelandic, Old Norwegian, Old Danish and Old Swedish. But the languages are not homogeneous - for instance there is as much variation within Sweden as between Swedish and Danish. Elfdalian and the Närpes dialect of Finland for instance, all retains features of Old Norse that Swedish does not. Elfdalian retains some things (nasal vowels) which no other Old Norse-descended languages do. Old Gutnish, spoken on the island Gotland in Sweden, was on a branch of its own altogether different from the East and West Norse branches.
The adage "a language is a dialect with an army and a navy" applies well here. You're reading a lot into the fact that Danish/Swedish/Norwegian are considered separate languages, but they're highly mutually intelligible today and were even more so at that time. And as stated, the dialectal variation within the countries was larger than that between the languages. There was also no official written standard for the languages until the Reformation.
In the 15th century, Denmark did not have a strong hold on the members of its union, in particular in Sweden after the Engelbrekt Rebellion. The king was mainly represented by his fogeder, who were mainly Danish and German. Low German was the lingua franca of the Baltic in that era (the zenith of the Hanseatic League) and had a huge influence on the Scandinavian languages at that time. But both Swedish and Norwegian did undergo some 'danification' in that era as well, due to the Danish administration. One example is Swedish 'bagare' for 'baker' where the Danish 'g' replaced the Swedish 'k', despite the verb 'baka' (to bake) still has a 'k' in Swedish. This was not the result of some conscious policy on the Danes' part though.
However the official 'standardization' of the language would come with the Reformation. I.e. just after Sweden had left the Kalmar Union, in the form of the first translation of the Bible into Swedish commissioned by king Gustav Vasa. Vasa hada an explicit agenda of standardizing the language and removing danisms from Swedish. The most visible change was Swedish adopting the forms "å, ä, ö" rather than "aa, æ, ø". The orthography was largely standardized as well - so there was a common written language no matter which dialect you spoke or how you actually pronounced the words.
In the case of Norway, which remained in union with Denmark, the reformation was largely a death-knell for Norwegian as a written language, in particular for administration, education and so forth. Danish became the standard written language, and for the next centuries Norwegians would speak their language but write Danish, and the spoken language became increasingly 'danified' as well. So its first in this era (1500-1800) that the real heavy influence on the Norwegian language comes, but by then it's the kingdom of Denmark-Norway and not the Kalmar Union.
(Sources: Bergman's Kortfattad svensk språkhistoria, Svenska Akademins Ordlista, Henriksson's Svensk Historia)