When I started playing classical guitars, I propped my left foot on a special foot stool and rested the guitar on my left leg, which was the standard position at the time. This angles the neck up at about 45%, which was good for accessing the fretboard past the money chords.
Later, that position become increasingly uncomfortable, and some of the classical guitarists I know were having problems with sciatica, and I looked around for a support that would allow me to keep both feet on the floor. There are several kinds -- pillows, leather thingees attached with suction cups or endpins, and I settled on the the latter, a Neck-up, from Hamre Music, in Vancouver, WA.
Works great, hasn't seemed to hurt the finish on the guitars, and while it would slip off now and again, I play sitting down, so the guitar didn't hit the floor. I swapped out the small suction cups for larger ones, and added a couple, and I'm pleased with it.
When I started playing ukulele, I had some trouble finding a position that would allow me to approximate the neck angle I liked. I can strum fine, but fingerpicking gets trickier the further up the neck I go, things start wobbling. Even though I play sitting down, I figured I'd get a strap, since one of my ukes came with strap buttons.
Then I checked with the Neck-up folks and found that they also make a mini-version, suitable for smaller instruments, like mandolins, narrow-bodied or children's guitars, and ta-dah! ukuleles, so I ordered one of those.
Forty-three bucks, shipping included. It does the trick for me. It will prop on either leg, though it doesn't help if you play standing. Might be something to look into if your lap seems a bit roomy for your uke. You can roll the strap up enough to get it into a large uke case under the headstock, or carry it in your gig bag.