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.

Related Questions

Swollen lymph nodes wch wnt down 70% aftr 1 week of medication with antiboitics, next day I had alcohol nd then immediately they were swollen again?

See below. If you are rubbing or palpating these regularly that cam cause them to feel enlarged. Leave them alone for at least two weeks and if still present see your doctor. Read more...