In Android Telegram, you can now Share a file with other apps. This means you can use a file explorer, e.g. ES File Explorer to save files to anywhere in the Android filesystem, including saving to the SD card.

File Save Steps

  1. Download e.g. ES File Explorer
  2. Tap on the download button on the file to download the file to cache first
  3. Tap to the side of the file to open the menu
  4. Tap Share
  5. Tap "ES Save to…"
  6. Choose save path
  7. Tap on Select button at bottom right to save.