We don't have any idea how the rollout of the Apple Intelligence features will figure in the betas. None.
Adding to the uncertainty is that WWDC was held a week later than its normal positioning this year. Will that mean a normally timed second beta (after 15 or 16 days)? A later second beta (skipping over July 4th week here in the US)? None of these options would surprise me.
The one thing that is 100% certain is that Apple needs to have new iPhones on sale prior to the beginning of their annual big quarter (10/1 - 12/31). So everything works back from that.
If I had to guess, I would say that we will get beta 2 on Wednesday next week (June 26). That will not be the version that becomes the public beta. That will occur with beta 3, which will likely come on July 9 (for developers) and July 11 (for the public). There will be no Apple Intelligence features in those betas.
If Apple is going to introduce any Apple Intelligence features for the iOS 18 launch (as teased by Mark Gurman), it will be the summarization/writing improvement type features. I suspect that we might start seeing those in DP 4 in mid-to-late July. But I also would not be terribly surprised to see those slip into 18.1/18.2 either, meaning later in the year. That would put more Apple Intelligence capable phones in the hands of consumers, softening the blow somewhat regarding the unavailability of these features for earlier phones.