Apple intelligence is their own project, with their own models. They bought a company specialized in AI at the edge (or on device). All the "AI" that will interact with user's data is Apple's.
Still waiting to see what the chat-gpt integration is exactly, but the more I read about it the more it seems it just the usual writing assistant that we will soon find in every text and image editor. And the hint that they will offer Gemini or other model as well really mean they haven't tied any real features to it.
According to the keynote at least, the integration is literally just Siri offering to defer to ChatGPT for some requests. Basically a more advanced version of "here's what I found on the web" if it doesn't know what to do otherwise.
Funnily enough, Apple isn't even paying OpenAI for that, they're literally saying it's for exposure.