Singing and strumming at the same time is very hard at the beginning.
One approach mentioned above is to strum only, until it's automatic. I usually advise to hum over that first, rather than sing full lyrics - it helps keeping to focus on the hands.
An entirely different approach, which can be used for variation, is to sing without a ukulele but 'hear' the strum in your head. It's easier than it sounds, like singing along with a headphone on. don't try even making hand gestures, at most use a modest nod while singing. The more you do that, the easier it will become once you pick up a real ukulele.
Perhaps this remark is off-topic, but I don't like it when a certain strum pattern is used from beginning to end in a song, even if it's a nice one like the one you mentioned. Loosen up, change pace midway, work your wrist and fingers instead of your elbow, don't be afraid to drop strum pauzes on dramatic bits, mark the differences between verses, choruses and bridges with your strumming hand... there's so much strumming expression that ukulele players have at their disposal!
