Is there an automated method of migrating all images from a wordpress blog to Flickr or Picasa, and updating the URLs in the posts they appear in ?

Firstly You need to download all your Photos from wordpress,

For that go to your host’s file manager,
browse to public_html > wp-content > Uploads >
select all folders and images and click on compress to zip. (every host gives compress option)
get all photos in bulk uploaded to flickr.
Now install the following plugin http://wordpress.org/extend/plugins/akwpuploader-alternative-wordpress-image-uploader/.
Go to your posts and delete the image (which was embedded from your own host)
then with this plugin you may add the image from your flickr account

This method requires some time and patience but there is no other solution.
But Once you complete editing all your past posts, its really very easy for you too create new posts without any extra efforts and you can easily embed any image from your flickr.

(dont forget to delete the photos from your wordpress folder to gain extra space!)

If anything you want to further discuss just comment.