Yikes
That's not good.
In the short term (next few days) there isn't much you can do other than protect it and keep it as clean as possible. The antiseptic cream is a good idea, I'd get some finger wrap too to protect it as much as possible. It's not sticky like plasters and is thicker so will provide some padding.
In a few days when the skin hardens and it's not quite so sore, and as long as it's not bleeding or weeping you could try putting a coat of nail polish over it, just for a little more protection. I'd go clear so you can see if anything happens underneath, but ultimately you'd be better leaving it alone completely.
It's going to take a good 4-6 months to fully grow out. Longer term damage is unknown until it grows, there is the possibility the nail bed has been damaged and the nail will never grow properly, it may always be week in that area or it may grow bumpy from the damage below. With some good fortune it will be fine, my Husband trapped his finger a few years back and lost the nail completely after it going black and falling off, however the new one has grown back perfectly fine with no issues. If you look closely you can see a slight discolouration on his nail bed which I think is scaring but the nail is perfect. It's really individual and no-one can tell until it grows.
