Xiaomi notification delay problem often occurs particularly on MiUI8, and are usually caused by the battery saver feature.
Xiaomi notification delay will recover quickly if the battery saver feature on some applications are not in use. I tried to apply this method on several types, such as Mi4i, Mi5, Redmi 2, Redmi 3S, Redmi Note 4, and I think it could be applied to all.
If back to MiUI7, Xiaomi notification delay occurs usually because the autostart option at each application. If it is not turned on, then the application will not run so no notification appears. On MiUI8 two interrelated causes that are autorun and battery saver. So you need to make the app Autorun-able and no restriction related with battery saver.
On the official forum Xiaomi also much discussed regarding this matter.
I write this article to facilitate answered questions of my friends who ask how to solve this problem more easily (without seeking again), through Youtube video tutorial below.
Below the Youtube Video Tutorial