Fill a large pot with lightly salted water; bring to a boil. Cook meatballs, working in batches of necessary, in the boiling water until they begin floating on top and are heated through, about 3 minutes. Remove meatballs with a slotted spoon, drain in a strainer, and transfer to a large saucepan.
Combine ketchup, apple jelly, red currant jelly, and hot sauce in a separate large saucepan; cook and stir over low heat until jelly is dissolved and sauce is heated through, 5 to 10 minutes. Pour sauce over meatballs. Simmer, stirring occasionally, over medium heat until meatballs and sauce are thoroughly heated, 15 to 20 minutes.