Whether healing is in the atonement is clear in terms of Isaiah 53: it is. But when that healing takes place is a matter of great debate.
There are many people who claim this passage to mean that because the atonement has already been rendered by Christ and He’s offered that healing, then we should not, as Christians, ever suffer any kind of affliction or sickness. That is on a collision course with everything the New Testament teaches about the suffering of the saints in this life, concerning both disease and affliction, prior to glory.
When we get to glory, it will be the end of all disease, all illness, and all death, which is one of the consequences of the perfect atonement that Jesus made.