TLDR: Google has an automated process with seemingly no manual review, and will blindly remove an app from worldwide sale in some circumstances (they seem to vary) if a trademark violation notice is filed with them.
As an app developer (not a lawyer), I'm working through this at the moment. A Polish company owns a trademark in the UK for a common English word (type of puzzle) which is in my app name. The USA store name is displayed in the UK app store (by default) and the holder filed a trademark dispute notification with Google.
Within 5 days, Google had removed the app from the store worldwide, even though the dispute was only for UK. Even though I had (within 1 day) already created a variation of my app listing for the UK, removed the offending word, and, received a confirmation email from the holder that I was no longer in violation, it was still removed by Google.
2 months later, I was finally able to restore the app by working through Google's byzantine appeal choices to find one where I essentially did an electronic "statutory declaration" that I was not in violation. (My assumption is that somehow this allows Google to reinstate the app in some automated way by punting it back to the trademark holder to resolve directly with me, but I really don't know.)
Postscript: Now the same company has found another puzzle game name to claim against me, but this time only in Poland. This time Google automatically stopped selling my app in Poland when they notified me of the claim. (However, now they've added France to the list and I see they've tried and so far failed to trademark the name in USA, UK & Australia.). It is so tedious trying to deal with all this.
If Google's implementation of this was reasonable, it would be relatively easy to create separate listings for offending countries and/or simply remove the app from sale for those countries. However my experience so far is that it is hit and miss, so I have reluctantly done what I view is the worst outcome and for now have adjusted my USA listing not to reference that trademark in any way just for convenience with an app that has few sales overall anyway. This lessens the possibility that they'll again disable my app for worldwide sales for 2 months. A chilling effect …
(So far, I haven't yet experienced this enforcement on the Apple side, but the trademark holder has just sent me an email directly to inquire about my identical app there …)