All the symptoms you have mentioned points towards infection around nail and nail bed. If there is pus accumulation we need to drain it to relieve pressure and pain. Plus you need a proper antibiotic course as per your weight and type of infection. You can use warm compress for pain relief.
Next Steps
Avoid putting water till its cured and also avoid dettol or any home based remedies.
Health Tips
Consult a qualified dermatologist for proper treatment