Hello, my name is***** and I have over 20 years of experience as a veterinarian.
I'm sorry to hear that your fellow Syd has a toe that is inflamed and swollen at the base of the nail.
Most of the time with only one nail/toe affected we are dealing with some sort of trauma (catching the nail and pulling it or twisting it) and a secondary infection or in some cases the pup may have a nail bed tumor developing.
Ideally he would see his veterinarian for an examination, possibly a radiograph of the affected toe and some oral antibiotics.
If you'd like you can try treating this at home for a few days and see how he comes along first, but only if the nail isn't loose. If the nail is loose it will need to be removed, so he will need to see his veterinarian.
He needs to stop licking the toe/nailbed so I recommend placing an e-collar (cone of shame, lampshade collar). You can buy one at most pet stores.
Soak his affected toe/foot in an epsom salt bath for 10 minutes at a time 3-4 times a day to decrease swelling and inflammation. Then rinse his foot and pat it dry. His foot should be covered with a sock and then a plastic bag over it, secured at the top with masking tape, so it doesn't get contaminated outdoors. It can be uncovered in doors.
If this is going to work his toe should look better in 48 hours and much better in 4 days or so. If it isn't then he should see his veterinarian.
Please let me know if you have any further questions.