When I use FileSavePicker.PickSaveFileAndContinue method in the visual, there is an error “ The FileSavePicker.PickSaveFileAndContinue() is obsolete: instead, use PickSaveFileAsync() ”. So you can use FileSavePicker.PickSaveFileAsync() method in Windows 10 mobile.

Update:
I have test Windows Phone 8.1 on Windows 10 Mobile, it was ok. The code of my project below , you can refer to.
You can also refer to this sample about FileSavePicker.
private void SaveFileButton_Click(object sender, RoutedEventArgs e)
{
// Clear previous returned file name, if it exists, between iterations of this scenario
OutputTextBlock.Text = "";
FileSavePicker savePicker = new FileSavePicker();
savePicker.SuggestedStartLocation = PickerLocationId.DocumentsLibrary;
// Dropdown of file types the user can save the file as
savePicker.FileTypeChoices.Add("Plain Text", new List<string>() { ".txt" });
// Default file name if the user does not type one in or select a file to replace
savePicker.SuggestedFileName = "New Document";
savePicker.PickSaveFileAndContinue();
}
/// <summary>
/// Handle the returned file from file picker
/// This method is triggered by ContinuationManager based on ActivationKind
/// </summary>
/// <param name="args">File save picker continuation activation argment. It cantains the file user selected with file save picker </param>
public async void ContinueFileSavePicker(FileSavePickerContinuationEventArgs args)
{
StorageFile file = args.File;
if (file != null)
{
// Prevent updates to the remote version of the file until we finish making changes and call CompleteUpdatesAsync.
CachedFileManager.DeferUpdates(file);
// write to file
await FileIO.WriteTextAsync(file, file.Name);
// Let Windows know that we're finished changing the file so the other app can update the remote version of the file.
// Completing updates may require Windows to ask for user input.
FileUpdateStatus status = await CachedFileManager.CompleteUpdatesAsync(file);
if (status == FileUpdateStatus.Complete)
{
OutputTextBlock.Text = "File " + file.Name + " was saved.";
}
else
{
OutputTextBlock.Text = "File " + file.Name + " couldn't be saved.";
}
}
else
{
OutputTextBlock.Text = "Operation cancelled.";
}
}