Typically, when you lose an asset such as in a hurricane, fire or other disaster and the insurance pays out, you simply report the "sale" of that asset to the insurance company for whatever they paid you. Then you enter the new replacement as it's own separate asset with a cost basis of what you actually paid for it. That cost basis would include both the insurance payout, as well as your deductible/out of pocket amount you paid.
However, that's not always possible in the case of losing the roof of a rental property, since the roof itself may not be listed as a separate asset from the structure. Then it's not possible to reduce your cost basis of the asset by the amount of the loss without really messing things up. In that case, you simply add the new roof as a physically separate asset and your cost basis would be your out of pocket costs, not including what the insurance paid out. Now if it costs you less than what insurance paid you, then the difference is taxable rental income and for simplicity, can be included in the rental income. But I don't see that as a possibility in these inflationary times.
Depreciation on the newly entered roof asset would begin once the new roof/property is placed in service. Typically this would be the day the new roof installation was completed, assuming the property remained classified as a rental during the time it took to install the new roof.
