Yes, that is a very good point. I have also thought about that sometimes, but since one half thinks about Westeros as a version of Europe and half thinks about it as a version of Britain, and since it's written from an English-speaking English/American perspective, one doesn't really think about that.
It would make sense for at least the North and possibly the Iron Islands to speak the Old Tongue, since they are First Men. I guess that somehow the Andals' language, the Common Tongue, slowly and gradually took over all across Westeros, but yes, precisely, that really doesn't make much sense. There isn't even that much trade between The North and the other kingdoms. But maybe the few great all-Westerosi things such as the Night's Watch and the maesters actually are a decent reason for that. If the maesters all were trained in Oldtown, the center for education of the entire contintent, like some specific universities were in Medieval Europe, then the Common Tongue of the Reach would have spread with the maesters and if the lords relied heavily on their maesters, then over time they would begin to speak more of the common tongue with their maesters, and then with the lower servants and smallfolk of the keep, and so forth... But yes, it's still strange.
A reasonable version of the three or four different languages across Westeros, like you mention, based on its history and geography, might look something like this:
The North - The Old Tongue/The Northern Tongue. (The "true"/most original version of the Old Tongue, since it has been in isolation the most and retained the culture of the First Men.
The Iron Islands - The Iron Tongue. (A slightly different version/accent of the Old Tongue, somewhat influenced by the Andals that came into the Iron Islands, maybe with some few specific loan words from thralls and so forth.)
The South (The Vale, The Riverlands, The Westerlands, The Reach, The Stormlands) - The Common Tongue. (The Andals' language, with some distinct accents/dialects for each kingdom, but still similar enough to be considered one single language. This is due to significant trade and the lack of strong geographical boundaries, the center of education around Oldtown in The Reach and so forth.)
The Hill Tribes of the Vale - The Mountain/Hill Tongue. (The First Men hill tribes, as the last isolated population of First Men remaining in the South, would probably have their own distinct dialect of the Old Tongue from several thousands and thousands of years ago and retaining it due to their relative isolation.)
Blackwater Bay - High Valyrian. (Not necessarily that impactful, due to the very small population of Valyrian noble houses, but still, considering the status of the language it might have spread throughout Dragonstone, Driftmark, Claw Isle and so forth, and especially in King's Landing if at least a few of the Targaryens would speak it to a certain degree at court and it could spread to courtiers, traders, dragonseeds and among the common smallfolk of the Blackwater.
Dorne - The Common Tongue & Rhoynish. (Maybe like 70 % of the population, especially the stony and sandy dornishmen speaking The Common Tongue, but many of the salty dornishmen still speaking Rhoynish.)
(Even within this, there would also be a few further specific languages and dialects within kingdoms and so forth.)