Tooth abscess and a swollen jaw, how long until antibiotics work?

See a dentist ASAP. If you rely on antibiotics or other not a definitive treatment, the pain always will come back. Dentist will treat a tooth pain by treating the cause and getting rid of the infection. Do not delay. See your dentist ASAP. Good luck.
Usually 2 days. Depending on the patient's health and the location of the abscess, there is usually a 2 day delay until improvement is noticeable.
By next day. By the next day. If not better, may be wrong antibiotic or abscess. If not better by next day, ask for referral to board certified oral & maxillofacial surgeon for assessment/treatment.
24 hours. Most antibiotic required 24 hours prior to reach therapeutic concentration at the site. Antibiotic should be the type accessible to bone (around the tooth apex) such as Clindamycin in the case of tooth abscess. Swollen soft tissue are best treated with amoxicillin.
Unknown. The tooth needs to be treated completely. Antibiotics alone will not resolve the issue without either root canal, extraction, periodontal therapy or combinations of treatment. Surgical drainage may even be required. I would see the appropriate specialist.