Do you know if the nails were only clipped or were filed using an electric grinder?
It sounds like the skin on the side of the toe above the actual nail is irritated. Is this correct?
Nail clipping alone, even if the nails are clipped too short and bleed, doesn't usually irritate the toe.
Electric grinders can get hot and burn. They can also touch the skin and abrade it. This does not usually happen, but is possible.
It is also very possible that the irritation is not related to the nail trim. He could have hit or caught the toe while playing, walked more than usual on pavement or hurt it some other way. The inflammation is most likely caused by his licking it.
Cold soaking will decrease the pain and inflammation. This can be done with a wet washcloth. Putting an ice cube in the cloth will keep it cold. Cold for 10-15 minutes repeated in an hour should help.
Infection is not usually a problem. Superficial infections usually self cure once the itching and licking stop. Peroxide is irritating to skin and is best avoided. The neosporin is fine, but he is probably licking it off.
Bitter apple spray, which tastes bad, often helps stop the licking. 1% hydrocortisone ointment (over the counter medication) helps if you can keep the pup from licking for a while. A Sock or bandage will also help keep the tongue off the area.
Let me know if you have follow up questions.