Its hard to see in the Burley photos, but the replacement photo shows a conventional wheelchair stub axle.
These are not particularly rare, but they're not bicycle parts. Instead, you'll need to look into medical or assisted living equipment providers.
Another option is to make contact with the equipment department at your local hospital.
Here's a closer photo of the wheelchair axles from my trailer.
two wheels, one showing inside and one showing outside.
Main axle housing, showing retainer clip.
As you can see, the main axle is a pipe/tube with a retainer clip. This engages a slot milled into the wheel's stub axle.
The axle itself is a piece of round bar stock with some grooves machined into it. There's a circlip holding the cartridge bearings into the wheelhub on the inside, and probably a simple bolt head on the outside.
In short, any competent machinist / lathe operator / fitter and turner could make these.
Mine do not have a press-button on the outside to release the wheel. It is just a solid cap.