Currently there is partial support for both the New York Times and the London times puzzles.

New York Times

NY Times uses the Across Lite format by Literate Software , which appears to have many different formats as well as some encryption. It seems that their recent puzzles are encrypted and older puzzles are in the clear. I am not yet sure whether they convert puzzles after a certain time or not. It looks like puzzles more than a week old are in the clear. If a puzzle is encrypted, it can still be imported to XWord, but the solution will not be available.

London Times

The London Times is an interactive puzzle and is simply described by an HTML page. Assuming their HTML layout does not change, puz2pil attempts to extract the puzzle from the HTML. You simply need to save the HTML document and run it through puz2pil.

Current restrictions:

More instructions on how to download London Times Puzzles.