This is almost exactly what happened to me in May. I had the tiniest little nick on my hand. At lunch I watched as a red welt appeared. I went to ER. In less than 24 hours my hand and arm were swollen just like you described. I was admitted for several days of IV antibiotics.
My doctor told me he had a patient who DIED from a paper cut in less than 48 hours. I was just lucky that I didn't wait to go to ER. My treatment (oral antibiotics first then IV) started with hours.
This happened to me on a Saturday. Had I waited until Monday to go see a doctor the outcome would not have been pretty.