Ok I've become a pretty big expert on barbed wire cuts now lol ;) as muss went through 3 electric fences, jumped a barbed wire and got caught in it after faceplanting.
Bute is your best friend. It'll keep them from being to sore and it really kept muss happy and calm. If he's in pain he may lose weight as my boy did, but you can always get that back. If theres an infection get a vet out asap! You don't want any infections getting worse, itll just make it longer to heal and can get worse quickly. Try clipping the area and washing it down with cold water twice a day, especially over the swelling parts and make it a light water running over, not a big rush of water if that makes sense. Keep him in a small paddock until it starts to get better so that he can't go too far. They don't look too bad heres a pic of muss's on the road to recovery
We used a yellow lotion spray on his cuts straight away which was really god and when they started to finally seal up we got a healing salve from Horseland tack shop that has worked wonders.
I hate barbed wire too but where I live if you find a place with wooden fences then its not an aggistment place lol