What causes instant swelling around cut?

I recommend going to an urgent care facility or see your primary care physician.  If you need a referral to a plastic surgeon, they will be able to facilitate that referral for you.  Please to not wait because this can cause scaring and damage that can be avoided by proper care and management.

Please seek medical attention for your injury. The swelling may be related to hematoma formation and/or active infection. It this is a foreign body injury - you may have retained foreign material in the wound. Appropriate management will spare you from scarring and additional procedures to correct a poor outcome from neglect.