Don't give up before you really try!
There are no "secrets." All it takes is work. I admit, sometimes a piece will flow naturally and write itself in an hour, but most of the time this is not the case. Expect to sit with a piece for days, weeks even.
Here's a couple of things you can try:
Most songs are written from one melodic motif. Take one of your snippets and record yourself playing it over and over. Let your fingers explore other avenues based on the snippet. Record everything. If your snippet is based on one chord, try it on other chords in the same key. After you've played for at least 15 minutes, listen back and see if there's anything you can use.
Perhaps two or more of your snippets can be combined into one piece. A bit of tweaking here and there may help you achieve this. Switch the key of one snippet to match another.
Sometimes the best stuff comes to you when you're not playing! If you're at your wit's end, put the ukulele down and hum your snippet or motif to yourself. It doesn't even have to be out loud, you can just hum it in your head. Where does the melody logically take you? Once you have something in your head, find it on your uke!
Hope this helps. Ask any questions you may have, Julie. Good luck and don't lose hope!