I find that 'eventually' my incomplete files will start downloading at some point. Though that might not be for weeks or months in some cases.

Check your settings, Tools -> Options -> Saving, Days to Keep incomplete files. Make sure this is a high number. Use a number like 9999 or similar.

When turning off LW, make sure it is totally turned off before turning off computer. Sometimes LW can be slow closing down (whilst writing to the prefs folder & stopping connections) & this corrupts the preferences ... thus loading problem for incomplete files when next starting up LW. ie: perhaps LW is not visible but is still closing down in background. Control-alt-del to bring up active programs will show you if it is still active. Perhaps even use the Disconnect feature from File menu for a couple minutes before closing LW. This is to help prevent this happening again.