Oh words! Fun!
I haven't actually studied the map yet, buttttttttttttttt... your breakdown seems logical, as far as naming progression
It will probably depend a lot on a few things though.
(Disclaimer: I'm NOT a linguist.. this is all just fun for me )
1) Scale/size/intent (they're tied in together).
First of all, in a smaller area, there are (for communication's sake) usually going to be less variation in dialects. You have to speak the same language as your neighbor to understand he wants your cow in exchange for his beans. The larger the area, the more likely that the language will diverge.
Ex: A village finds a glass bottle (stole that idea from a movie) and they have never seen anything like that before. Everyone is going to look at it and play with it, but if Big Chief calls it a "Yutime" then everyone is going to call it "yutime".
If glass bottles find their way into a few nearby villages, who trade with each other often, then they will, similarly, attempt to find a common name. But it starts to break down from this point on.
Big Chief of another village might call it something else, and they will either blend the names somehow, adopt a single name for the bottle, or continue calling it different things.
Someone from a village with no bottles comes and hears this strange word "Yutime" one time, as he trades his magic beans for it. He takes it home and can't pronounce yutime so he tells his friends it's a "Yuthim".
Then you get into common terms and slang, and...
And.. lots of stuff... point is.. the further apart the people are, the more diverse their language becomes.
If all these different conquerors lived on ONE smallish continent, and no one came from elsewhere, their languages are likely to be more similar, overall.
If you look at it, European nations tend to have similar languages, and african languages follow a certain pattern (except for french - which has its' own history) and germanic/dutch/nordic languages... they're all different, but similarities tend to grow stronger the nearer they are to one another.
(I repeat: LOTS of generalizations here!)
Given that it's a large/diverse enough area, and your names work that way... you mentioned "anglicized names" for certain areas, and that's true. But people who live in those cities will not, typically, call them by the anglicized names. So if someone outside your mapped-area *(or from one region with a totally different language/dialect) is making the map, you might see more diversity in the names, as well.
2) How long did the conquerors remain? It usually takes anywhere from 10 years to a couple generations for people to adopt the language of a conqueror. If I take over your country, and tell you an apple is called a "hapr", you might say "hapr" in front of me, but at home, or with your neighbors at the store, you're going to say "apple". It takes the *children* learning "hapr" to change the name of the apple.
So if your kingdoms were overrun by several armies in quick succession, all the apples (and the cities) will still have their common names.
If the militaries stayed a while... then yes, it makes sense that many things would have changed.
3) What do YOU intend for your map?
There's a difference between accuracy and what people (in general) will accept as accurate.
My neighbor SWEARS there are 58 states in the USA (if you're not from here and don't know, there's 50). She cannot accept that there's 50, no matter how much I do to prove it. It just *feels* wrong to her.
If the names *feel* wrong to all your players, or readers, or w/e, and they aren't the type to fact-check and adjust their opinions, then they won't ever accept this discrepancy in the names.
If they're more open minded, they might.
If you don't have an intended market (if it's just for you) and strict accuracy matters to you, or if it's for some sort of academic project, then do it up right.
Unfortunately, you can't really know how others will react until it's a done thing, but...
Anyway... I'm going to shut up now, and examine your map/history, because it keeps me entertained