Here is what I did after speaking chatting with Apple. They said they are working on a fix and no more new tickets about this issue are being written up. The engineers, I am told have enough information and will have a resolution soon.
What I did: (Original update to iOS 9.3 was done over the air without iTunes)
1. Made a current , right now back up with iCloud. I was going to do a complete phone erase and restore.
2. Plugged into my PC and went to my iPhone details. Restore iPhone.
3. When the software download completed, iPhone will delete all information and ask if you want to setup from a previous iCloud or iTunes backup, or restore as a new iPhone. I chose to 'restore as a new iPhone' By doing this, it would allow me to test Safari, Mail and iMessage functionality. I wanted to see if the links worked as they should without any of my apps or old settings. A new phone out of the box.
4. Checked out all software and it checked out OK. Everything worked as it should have!
5. Now I did a erase phone and settings from the iPhone, without plugging into iTunes. I went through the entire process and when asked again about where to restore from, I chose to 'restore from iCloud backup'.
6. Checked functionalito of said problem Apps, and much to my amazement, everything worked as usual.
Now I will monitor this and see if it holds. I honestly don't remember having this issue on Tuesday after the announcement having this issue. I think it started a couple days later and really not all at once. I use Mail links every day and News as well. Seemed fine after install. I don't know. I do know now the issue has been fixed on my iPhone 6s Plus. Now to fix my girlfriends 6s Plus the same way. Both of our phones were affected, but oddly enough, the iPad pro was not affected at all with the 9.3 update.
Hope this works for others. It is frustrating to have the 'most advanced software on the planet' screw up on us like this. Only for some, not all. Bizarre.
Ray