Save video to the gallery and get the path for the video stored to Gallery

This worked for me perfectly.

Swift 3.1 ->

PHPhotoLibrary.shared().performChanges({
            PHAssetChangeRequest.creationRequestForAssetFromVideo(atFileURL: url!)
}) { saved, error in
            if saved {
                let fetchOptions = PHFetchOptions()
                fetchOptions.sortDescriptors = [NSSortDescriptor(key: "creationDate", ascending: true)]

                // After uploading we fetch the PHAsset for most recent video and then get its current location url

                let fetchResult = PHAsset.fetchAssets(with: .video, options: fetchOptions).lastObject 
                PHImageManager().requestAVAsset(forVideo: fetchResult!, options: nil, resultHandler: { (avurlAsset, audioMix, dict) in
                    let newObj = avurlAsset as! AVURLAsset
                    print(newObj.url) 
                    // This is the URL we need now to access the video from gallery directly.
                    })
            }
}

First, you need to set the following permission in your app's plist file:

Privacy - Photo Library Usage Description

Provide a string that is presented to the user explaining why you are requesting the permission.

Next,

import photos

Then use this code to store video

    PHPhotoLibrary.shared().performChanges({
        PHAssetChangeRequest.creationRequestForAssetFromVideo(atFileURL: fileURL)
    }) { saved, error in
        if saved {
            let fetchOptions = PHFetchOptions()
            fetchOptions.sortDescriptors = [NSSortDescriptor(key: "creationDate", ascending: false)]

            let fetchResult = PHAsset.fetchAssets(with: .video, options: fetchOptions).firstObject
            // fetchResult is your latest video PHAsset
            // To fetch latest image  replace .video with .image
        }
    }

To get url from PHAsset, refer to this question's answer

If you want to fetch the saved element (not using a sort descriptor with date order), you can do it this way:

var changeRequest: PHAssetChangeRequest?
var blockPlaceholder: PHObjectPlaceholder?

PHPhotoLibrary.shared().performChanges({
            changeRequest = PHAssetChangeRequest.creationRequestForAssetFromVideo(atFileURL: url!)
            blockPlaceholder = changeRequest?.placeholderForCreatedAsset
}) { saved, error in
            if saved {
                guard let placeholder = blockPlaceholder else {
                    return
                }
                let fetchOptions = PHFetchOptions()
                let fetchResult:PHFetchResult = PHAsset.fetchAssets(withLocalIdentifiers: [placeholder.localIdentifier], options: fetchOptions)
                if let asset = fetchResult.firstObject {
                    //here you have the PHAsset
                } 
            }
}