Time warp is often used by the pros to prevent kiting, delay a two pronged attack or just literally delay a whole attack. Remember that TimeWarp only affects movement speed, it does not affect attack speed. The basic of protoss early and mid game engagements stays the same: engage in choke points to abuse your ffs and not be overwhelmed as easily.
Time warp AoE is large enough to affect most of a 3 ff wide ramp. It can help make the units come up slower (stimmed marines, speed roaches) and makes ffs easier on units in the Time Warp. It also make kiting harder for range units like the marines and marauder. Used correctly, it will allow your zealots to get the 3-4 additional attacks that can make the difference between losing a battle or not losing a battle.
If a two pronged attack is about to hit your middle game army, dropping the Time warp on one of the push will give you more time to attack the opponents army the way you want. It can also force the opponent to go where you want him to go since engaging anytime in a time warp is a bad idea because of the movement slow.
If you see a big army coming your way, dropping the TimeWarp will delay this army since, like I said before, engaging in a timewarp in the middle game where the Protoss has either colossus or templars is a bad idea. It might also allow you to drop money storms during a battle since dodging a storm in time warp is close to impossible.
These are the things I've seen WhiteRa, Grubby and Huk do with the TimeWarp.
Also just remembered, TimeWarp on the worker line slows mineral income, funny little use for it if your momma core ends up behind an enemy line.
Allows your nexus to become a cannon for a limited amount of time. It will make 1 gateway fe easier to pull off. The photon overcharge gives your nexus the attack of a cannon and the range of a siege tank (or a little bit more than a siege tank). It allows for quick defense and makes the use of a cannon less of a must while early expanding.
Also, a troll tactic of doing a planetary nexus in your opponents main can work if done right and your opponent is in the lower leagues.
It is best used for hit and run type of attacks with your army. It allows you to teleport after a small delay to any nexus on the map. Perfect for hitting the zerg third and then defending your own third. Or moving fast across the map in the late game when you have 5-6 bases.
EDIT: Just noticed that you might want all of the uses of the momma core abilities, so I added more.