Hello from JustAnswer.
The flucloxacillin is a good antibiotic for infections of the skin and skin structures, such as an infected sebaceous cyst.
By the fourth day of antibiotics, there should be some definite improvement if the antibiotic alone will be sufficient for treatment. The symptoms may not be completely gone by this point, but there should be definite improvement. It also should not be such subtle improvement that you are having to wonder whether there is definite improvement. As long as there is this definite improvement, then it is fine to give the antibiotics a few more days.
In many cases of infected sebaceous cysts, though, it may be necessary to drain the cyst. And drainage would typically require cutting into the cyst, not just sticking a needle into it. If there is not the definite improvement with antibiotics and the cyst has not been drained, it would be appropriate to contact your doctor for drainage. Or if you are still not at home, you can contact an Urgent Care Center or Walk-In Clinic for consideration of drainage.
If I can provide any additional information, please let me know.
Their is no need to be referred to a specialist. This is a very minor procedure that can usually be done in your GP's office.