HaroldvandeKamp, the locations you mention (the Copilot app in Teams, the Copilot app for mobile, Microsoft365.com and Copilot.microsoft.com) are essentially all one and the same. Initially they weren't but Microsoft made an effort to make them the same, also in look and feel. Therefore they also have the same Copilot prompt/response history, so you can continue at any of those locations with new and earlier prompts, indeed in the Dutch language.
With regard to our Dutch language in M365 Apps (read the traditional Office Apps)..., I guess we need to try it out every day to see if it is active in the tenant we each work in. I suspect we don't need a specific channel build, although when we spoke you thought otherwise.
Update May 3, 2024:
I confirm I can use the Dutch language in Word in my employers Tenant using Current Channel build 2403/2404 on April 30th, 2024
The next day (May 1, 2024) it also worked in the Tenant of the customer I work for using Current Channel build 2404.
In PowerPoint it does work right now in current channel 2404 but not in monthly enterprise 2402. It will take a while before it is available in all M365 (Office) Apps and update channels.