Getting a trademark doesn't actually ensure you the sole right to the name, unless you have a market reach to justify you owning that name. If you have a published game that predates any registered trademark you should be able to convince most courts that they can in fact not force you to change the name. Unless of course you choose a name close to an existing brand. Or Google/Apple decide that you are not important enough.
King.com tried and failed to trademark Saga, the Escalator company lost the rights to their brand, as it simply just became the word for moving stairs. Same thing happened to CocaCola in Denmark where they lost the monopoly on the word cola, because it just became the name of that particular kind of soft drink. Generally brands can be disputed if no one can connect it with a specific company/product. But this is a HUGE legal field spread out in several independent legal instances across the globe.
In short worry about getting your brand established first, unless you have an army of lawyers ready in every region of the world.
Also I have no education in any field close to this, I'm just saying what I've heard at various business talks