Thanks, guys, like you say "it depends". But those pointers allowed we to find the answer for my own case which is:
"Yes, if I get 3/8 inch BMX axles and bend (cold set) the rear frame or add spacers."
Cheap store-bought bikes like mine have 3/8" or sometimes 10mm axles, which are practically the same. BMX ones are either 3/8", or are 14mm, which is a good bit larger.
It's not quite game-over for the 14mm BMX axles though because some have flats on them so they fit 3/8 dropouts as well. If the axles are solid you can file the flats yourself - but a lot are not.
Opinion was pretty unanimous - do NOT file-out the dropouts to take he larger axle.
The width of the road and the BMX axles are both generally 100mm so the 3/8" BMX front wheel often drops straight in.
The rear axle is significantly wider on a road bike - 130mm on the road vs 110mm on a BMX. The BMX axles might be over-length enough to fit if you put in a 10mm spacer each side. Or you can bend-in the rear frame if it is steel to take the narrower axle (cold setting).
Just want to reinforce that the full answer is indeed "it depends", and this answer is specific to for cheap store-bought bikes with steel frames.