The nail hole could still be sore, just like if you ever jabbed something under your nail by accident, your finger feels the effects for days afterwards.
Even though the vet flushed it once, it could still become infected. As a preventative, I would recommend soaking in epsom salts and warm water for about 10-15 minutes a day if possible to help draw any infection out . He may still develop an abscess, which can take a while to form and it hurts worse before it gets better. The best thing to do is wait it out, if he's in obvious pain, let him rest, but don't confine him in a stall if you don't have to. Movement helps the body deal with the wound and any abscesses resulting from it by increasing circulation. Confinement would just drag it out longer. If it abscesses, he may show no further symptoms than he does now, or he could act like his leg is broken.
Once the abscess "pops" like a pimple (pus oozes and can be green, yellow, white or even black and come out at the hairline or anywhere else in the hoof in line with that hole, including the sole) Don't have it cut out, it can create more opportunity for a worse infection. Just soak it and let it drain. When it pops open, soak it to clean it and put some salve on it like Ichtamamol on it. His lameness will probabl seem to magically disappear once it's popped
Give it a few days befoe working on any real training, regardless. Infection or not, the foot could be tender and having the farrier use the testers to squeeze the foot can be inconclusive.Watch for heat in the foot (compare to the other front foot) and light riding can be continued if he's not too sore, but if he does show apparent lameness, stay off him until it's over