your on the right track with the neosporin.I would get some epsom salts and get the dog to lay down and let it soak.you can get better results usually if you lay with the dog first,getting it comfortable.
soak ir 3-4 times a day for 10 minutes,starting with warm warm water,disolve the salts.then apply the neosporin.
if the nails split you'll have to probably get antibiotics,keep it clean and washed regularly.
we used to have black drawing salve in the way back we called it mule grease.it smells almost like diesel fuel.it would draw the infection out.
you can also make an Allum paste with powder and water and apply it.that was another thing we would do.it comes in small bottles.
you may get it under control if you've got a 3 day wait for the vet.
remember,keep it clean.and don't use peroxide.
you can wipe it down with alcohol.
if it looks like it's clearing up afte a day or so.after washing it use betadine and let ir air dry.
theirs all sorts of ways to do it.these are some of the ways we would take care of our dogs,and we always had banged up dogs.