How long is it supposed to it take for antibiotics to work on swollen lymph nodes?

Depends: If the swollen lymph nodes are from a bacterial infection, then you should notice an improvement in 2-3 days. If they are from a viral infection, antibiotics won't help. And if they are due to another cause and not an infection, antibiotics won't help either.

Monitoring Centers: Lymphatic system absorbs/carries things too big to go into the arteries/veins like bacteria eaten by white cells and digested food. Lymph nodes "taste" the lymph for anything bad like bacteria/viruses/cancers, etc. When it detects a problem, it alerts the immune system and your body reacts. They enlarge in response to the inflammation they create when they react. They shrink when all is well.