What should be done to atone for the sin of my wife undergoing a C-section in a free government hospital, while I have the financial means to have it done in a private hospital? And is this considered a grave sin?

The wife must be taken to a hospital where women handle deliveries. However, if this is difficult due to prohibitively high costs that would cause financial harm, there is no sin in not doing so. But if the decision to avoid the private hospital is not due to a pressing necessity, and there is laxity in adhering to Islamic guidelines, then one is sinful for that and must repent.