Once a claim has been filed, you must wait until the warranty service authorizes repairs before any work can begin. With a reliable extended warranty provider, this shouldn't take more than three to five days. If a part isn't installed properly or the body panels start to rust too soon, these are issues that will be covered by the factory warranty. If you want the cost of a typical oil change to be covered by the warranty, you won't be lucky.
If the vehicle is under factory warranty, no approval process is necessary. The customer arrives with a complaint, the technician diagnoses it and fixes it. If it's related to the engine problem, they need to make a diagnosis and get engine approval, but it usually only takes a couple of days. Once a claim is approved, you can take your vehicle to an approved auto shop to do the job.
Unlike insurance that normally reimburses you for repair costs, you probably don't have to pay anything to an approved auto shop that works with the warranty provider.