Kid's bike - Can I put 20" wheel on a frame that was meant for 18" wheels? So, my friend's kid's bicycle has an 18" tire that is completely worn down. I'm having a really tough time finding 18" tires on local buy/sell websites, but there are a bunch of 16" and 20". Does anybody know if putting a 20" (or 16") wheel on my friend's kid's bike will work? No gears, just back pedal braking, and a hand brake for the front wheel.

Thanks a lot.
There are several different 18" sizes, and several different 16" sizes. See Sheldon Brown's tire sizing page

If the 18" is the 355mm used by Birdy, and the 16" is the 349mm used by Brompton, that's close enough that it might be possible.

People have converted Birdys from 18" to narrower 20" (406mm) by moving the brake posts (see "brakeboss relocator").

I changed the obsolete 500A (440mm) tyres (and chromed steel rims) on an old Dawes Kingpin folder to 20" (406mm) ones on alloy rims, the widest ones that would fit in the fork to reduce the bottom bracket drop, and was able to find suitable brakes to fit.

But for a typical child's bike, I very much doubt it's worth it.