Same issue here with iPhone 7 Plus updated from iOS 10.3.3 to iOS 11 and used with a Apple Watch Series 2 running Watch OS 4.
Have tried all of the steps supplied by Starr.C plus more with no positive results.
Same bug being all text, MMS or iMessages do not wake the screen on the iPhone when they are received by the iPhone, but the Apple Watch does alert me that a message has been received.
Have tried with silent switch enabled and disabled with no luck and found that when silent switch was disabled that there was no text tone either for when the message was received by the iPhone, so no waking of the screen display and no audible text tone.
But the funny thing is if I disable or turn off bluetooth everything works fine.??? Strange.
Incoming mail alerts work fine waking the screen using the default Mail app and also the Outlook app with Hotmail accounts.
I think this may be some sort of bug with the new feature of (DND) do not disturb while driving as I believe it has a feature or option where it can be enabled automatically when connected to a car bluetooth system. Not sure but possible I guess.???
All that I could say to BryC is to test the iPhone 7 with bluetooth disabled.
I also reset the Apple Watch and restored from a backup to see if it was the problem but it did not fix the issue, next step if it was the Apple Watch would be to restore as new but I am sure that this is a bug with the iPhone and iOS 11.
Early days with a new iOS with lots of wrinkles to iron out I guess.