A bit late to the party, but here I go:
You could save the TextGrid
object as a JSON file and read it into NLTK using the standard python libraries as in this answer.
Praat does not include (at this time) a JSON converter, but [full disclaimer] I have been working on one such script that should do the job. It's part of a larger plugin I maintain, which can be downloaded from its github repository.
Once you install the plugin, you can use it by running
runScript: preferencesDirectory$ + "/plugin_jjatools/save_as_json.praat",
..."/output/path", "Data stream", "Pretty printed"
That script basically calls a perl script in the background, which does most of the hard work, so you could also just run the perl script directly. Even though it is still in development, most object types are currently supported.