I would like to be able to hook it up to one of my mower 12 volt batterys I keep in the house during the winter...under maintenance charge.
You can get nebulizers that will run straight off 12 volt DC current.
You can get "jump starters" with built in inverters that will easily run it.
They also give you a flashlight and an air compressor.
You can also get small inverters that could connect directly to your battery.
I'd stay away from any hand cranked charging devices.
I've not seen any that are all that reliable.
You're only talking about around 20 watts draw.
Even these smallest ones will run the nebulizer:
https://www.harborfreight.com/automotive-motorcycle/power-inverters.html
How long it will run will be determined by the amp/hour rating of the battery.
I don't know how long a small mower battery will run your nebulizer. I doubt it will go for 24 hours.
Nebulizers are used to deliver medication in the form of a fine mist.
They normally only run for about 5 minutes each time the medication is needed.
Oxygen concentrators take a lot more power and are more likely to be run for hours on end.