If you do not have access to Epsom salt, then dipping your finger in warm water will not help. In fact, the opposite would probably work better, where you place your finger in extremely cold water, as this can relive swelling and inflammation for a while.
Inflammation after an injury is the standard response of the body, specifically the immune system. Inflammation occurs as a result of histamines and other substances secreted in the body that cause swelling. The swelling itself is for white blood cells to permeate the smaller capillaries and fight off any invading bacteria. Epsom salts have an effect of reducing inflammation because of its anti-inflammatory properties. The other natural anti-inflammatories include ginger and aloe vera. While aloe vera is quite effective against open wounds, ginger is especially effective when dealing with inflammation within the body. Ginger can be consumed along with tea or as a crushed supplement to your diet. You should inspect your injury to ensure that there is no external wound, especially near the area of the nail. These kinds of injuries can end up in an infection. Just to ensure that this is also not a possibility of the same taking place, apply some tea tree oil to the area. Tea tree oil is a very powerful antimicrobial that works against bacteria, viruses and fungi. There would also be a substantial amount of bruising in the area. Bruising occurs when the blood vessels of the area are damaged, and spill their contents into the surrounding tissue. This fluid accumulation must also be dealt with by ensuring that the finger is at a raised position when at rest. For example, when you are lying down, ensure that your hand is placed on a pillow to drain the fluid out of the surrounding tissues.
Using analgesics that are over the counter medications is the most effective in dealing with injuries of this kind. The simplest analgesic that you can use is one that contains acetylsalicylic acid or ibuprofen. Being analgesics, these reduce pain and also help to reduce swelling and inflammation. As the finger starts to heal, it also helps to start moving the digit as this improves the circulation in the area and consequently the healing process as well. Once the inflammation starts to subside, start heat applications to the area but at a temperature that is tolerable to your finger, as it is still tender from the injury.