I think the nail bed is infected after the previous injury
and only topical antibiotics will not help to cure the infection because very small quantity will actually reach to the infected area. In which case you need to have oral antibiotics
and that can only be obtained by a prescription from your GP.
The other thing is that the nail folds should be examined properly if there is an embedded toe nail? Which can cause the problems.
So I think untill you see your doctor you can use Tablet Advil
400 mg thrice a day after your main meals for 3-5 days to reduce atleast the inflammation going on.
Secondly you can use Dome borrow compresses to relive it.
Keep clean the area with only warm water with a little salt in it and then dry it.
If you get a chance then pleaese see a doctor who can have a look at the wound.
Thanks and all the best.