Note that the ukulele is cradled by the right arm, between your wrist/forearm and body. The index finger of the right hand is extended and is used to strum the strings, right at the place where the neck of the ukulele meets the body. Don't strum near the hole. The fingernail of the index finger faces the ground, so that the back of the nail strums the strings on the way down, then the tip of the finger and nail strum on the way back up. The ukulele is held high on your body. This position allows you to partly hold the ukulele up in the crook of your elbow. It also helps you to hear the ukulele while you are singing, when it is closer to your head. Your right forearm should point right up the ukulele's neck.
If you are holding the uke properly, you should be able to take either hand away and have the instrument stay where it is. That's easy enough for the left hand - grab the neck between the heel of your hand and your fingers, as though you're making a chord. It's a little trickier for the right hand - you have to keep the whole hand free for strumming, so you have to get used to sort of cradling/squishing it in there.
Just try it! It's worth getting used to hanging on with just the right arm and not the hand, otherwise when you are playing songs, the ukulele will keep catching you off guard by slipping.