Ever since I installed IoS 9.3 on my iPad 2, the passcode lock doesn't work unless Require Passcode is set to Immediately. Specifically, whether I let the screen go dark after the Auto-Lock time, or close the cover, if Require Passcode is set to something else, I never get a passcode prompt, while if it's set to Immediate, I do. Has anyone else seen this?
You might try to hold home and lock buttons for a while until the device reboots the OS. This startup will require a little longer because it will be deeper than normal restart.
iOS 9.3 has brought many bugs, this should be one of them. I believe for now you should lock it with code time set to "immediately" (for safety reasons), make a YouTube video to explain the issue (I did a similar thing for a buggy update of PlayStation Vita, and it got fixed), report it to Apple and wait for 9.3.1 to come out.