If a finger or toe has been cut off (gruesome, but it happens), apply pressure to the wound, then place a cloth, towel, or plastic bag around the body part and ice it. Make sure it isn’t directly touching the ice and is protected. The body part can probably be reattached if the surgery’s done within six hours.
From The Smart Parent's Guide: Getting Your Kids Through Checkups, Illnesses, and Accidents by Jennifer Trachtenberg.

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The Smart Parent's Guide: Getting Your Kids Through Checkups, Illnesses, and Accidents
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Important: This content reflects information from various individuals and organizations and may offer alternative or opposing points of view. It should not be used for medical advice, diagnosis or treatment. As always, you should consult with your healthcare provider about your specific health needs.