The club were hit with a Covid-19 outbreak in the build-up to last Sunday's game at Racing 92, forcing the Champions Cup clash to be cancelled.
Saints did not travel to France and shut their training ground as a precaution.
They underwent testing on Tuesday and were able to get back on the paddock after receiving the results on Wednesday.
And now the signs are good that they will be able to fulfil their Gallagher Premiership game at Sixways this weekend.
"We've had to miss a couple of training days but it hasn't had a huge effect on us and we're back in today," Vesty said.
"We'll be able to train for a couple of days before the game on Saturday.
"Most people who got it were pretty symptom-free and wouldn't have known they had got it had they not had the PCR test.
"We've just had to rearrange our training a little bit because we were planning to come back from Paris on the Monday so we would have trained Tuesday and had Wednesday off. We've just flipped that Tuesday and Wednesday around.
"It hasn't changed things massively.
"We've got a few people who are missing but due back tomorrow and it's all expected to be fine."
Vesty added: "We've had a pretty clean batch (of tests) come back and we're as confident as we can be that it's out of our environment.
"We lateral flow anyway to try to stay one step ahead and because we were proactive with staying away from the training ground and not mixing together, I think we've stopped it as well as we could have.
"It was really well managed from above and our medical department have done a cracking job to allow us to go forward for this next block.
"If we'd got this wrong and it ran rife, it would be very difficult to stop it then."