Possible nail fungal infection, finger red, swollen, painful, doctor gave me flucloxacillin, why? if its ineffective on fungal infections?

Probable bacterial: This sounds like a bacterial infection(paronychia) for which warm soaks and antibiotics may be helpful, but if it is a paronychia the nail often has to be incised a bit to let the pus come out. See your physician if the inflammation persists.

Poss bacterial infec: You may have a paroncyhia, which can be a bacterial infection. If the prescribed treatment doesn't help within about 2 days, or if it's getting worse, then i'd recommend you call the doctor who prescribed this and let them know. They may want to prescribe if different medication and/or see you back for follow-up.