Is it possible for strep or tonsillitis to still get worse the first day after starting antibiotics?

Possibly. Bacterial infections can worsen, even if beginning antibiotics. Sometime patients feel worse in 24 or so hours due to substances released into the body from the death of the bacteria and cell death. Antibiotics do not guarantee cure. If one is worsening after beginning antibiotics and there is concern, a phone call to the doctor may be warranted or a re-visit. Give it a chance then call.
Yes. It takes at least 48 hrs. For the antibiotic to reach the tonsils and start to work. Things should start feeling better by day 3.