iPhone now goes to sleep while watching videos

Correct me if I’m wrong but the iPad doesn’t have a low power mode, and if same issue is happening on that device it’s probably not behind the issue, did you do the updates via OTA, if so and you have access to a computer you could try restoring your devices via iTunes.

Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

Make sure you have a BackUp first though so you can select that option to get your data back, alternatively you could setup as “New” just to test things out and restore your BackUp later.

Done a quick search and issue appears quite common iPhone Sleeps While Watching videos | MacRumors Forums but still no reason found to why, so a bit hit and miss trying to resolve I’m afraid.

I've just checked my iPad mini 2 and iPhone X for this very same issue and they appear ok at the moment (on iOS 11.3.1) and iPhone is set to 2 minutes Auto Lock, video watched was on YouTube, so it doesn't appear to be something contained in the update itself, rather it's more likely come about from a series of changes that finally combine to cause the problem you are seeing.

I think my last update was done OTA for both devices but can't remember the previous ones, my view is any major update should be done via the Mac, but point updates I'll go either way, it's worked fine up to now but that could be down to just luck, who knows.

You can always try the OTA Restore method first if that is easier How to erase your iPhone, iPad, or iPod touch - Apple Support, you "Erase All Content and Settings" via your device prior to using iCloud to bring back your data (select "Restore from iCloud Backup" when at the relevant screen). But if the issue is not data related but contained in the OS itself brought about from a faulty update (or series of updates) this will probably not correct the issue. For that you will need to completely replace the OS and using iTunes achieves that goal.

Best of luck.