You can use flatMap or compactMap to do this like:
If you're using swift 4.1 use compactMap otherwise use flatMap.
// Your dictionary
let dictionary = response as? [String: Any]
// Create an empty Array
var myFileArray = [String]()
// Get the doc array
if let docs = dictionary?["doc"] as? [[String: Any]] {
    // for each doc array
    for doc in docs {
        // get the nameChange array
        if let nameChange = doc["nameChange"] as? [[String: Any]] {
            // convert the inner dictionary to string array
            let fileNameArray = nameChange.compactMap { (dictionary) -> String? in
                return dictionary["file"] as? String
            }
            // append the array into main array
            myFileArray.append(contentsOf: fileNameArray)
        }
    }
}
// Here is your final string array of filename
print(myFileArray)