I finally fixed the problem.
There doesn't seem to be any clear explanation here of what is happening, so I'll try to explain it as I understand it.
When some people upgrade (or perhaps do a fresh install) for some reason or another some crucial tables are not being properly updated. In my case, I did an upgrade, and three tables were not updated. I had to manually add insert new data to get my WP install to work.
All of the data that is missing relates to the new permission system. I strongly suspect that for people who skipped a version or two of WP, these tables are not being properly updated by upgrade.php, as I had no problem with WP installs where I have been doing regular updates. I think this is a bug and should be fixed.
There are two tables and three rows of data I had to manually insert.
The first table is the wp_usermeta table. You want it to look exactly like this (mine was empty!):
http://www.tamba2.org.uk/wordpress/admin/
The second table is the wp_options table. You need to add a new option called "wp_user_roles" which is not being created properly. There are only two things you need to change when you insert a new row into this table. First, "option_name" should be "wp_user_roles". Second, the "option_value" should be set to the following very long string of text:
a:5:{s:13:"administrator";a:2:{s:4:"name";s:13:"Administrator";s:12:"capabilities";a:30:{s:13:"switch_themes";b:1;s:11:"edit_themes";b:1;s:16:"activate_plugins";b:1;s:12:"edit_plugins";b:1;s:10:"edit_users";b:1;s:10:"edit_files";b:1;s:14:"manage_options";b:1;s:17:"moderate_comments";b:1;s:17:"manage_categories";b:1;s:12:"manage_links";b:1;s:12:"upload_files";b:1;s:6:"import";b:1;s:15:"unfiltered_html";b:1;s:10:"edit_posts";b:1;s:17:"edit_others_posts";b:1;s:20:"edit_published_posts";b:1;s:13:"publish_posts";b:1;s:10:"edit_pages";b:1;s:4:"read";b:1;s:8:"level_10";b:1;s:7:"level_9";b:1;s:7:"level_8";b:1;s:7:"level_7";b:1;s:7:"level_6";b:1;s:7:"level_5";b:1;s:7:"level_4";b:1;s:7:"level_3";b:1;s:7:"level_2";b:1;s:7:"level_1";b:1;s:7:"level_0";b:1;}}s:6:"editor";a:2:{s:4:"name";s:6:"Editor";s:12:"capabilities";a:19:{s:17:"moderate_comments";b:1;s:17:"manage_categories";b:1;s:12:"manage_links";b:1;s:12:"upload_files";b:1;s:15:"unfiltered_html";b:1;s:10:"edit_posts";b:1;s:17:"edit_others_posts";b:1;s:20:"edit_published_posts";b:1;s:13:"publish_posts";b:1;s:10:"edit_pages";b:1;s:4:"read";b:1;s:7:"level_7";b:1;s:7:"level_6";b:1;s:7:"level_5";b:1;s:7:"level_4";b:1;s:7:"level_3";b:1;s:7:"level_2";b:1;s:7:"level_1";b:1;s:7:"level_0";b:1;}}s:6:"author";a:2:{s:4:"name";s:6:"Author";s:12:"capabilities";a:8:{s:12:"upload_files";b:1;s:10:"edit_posts";b:1;s:20:"edit_published_posts";b:1;s:13:"publish_posts";b:1;s:4:"read";b:1;s:7:"level_2";b:1;s:7:"level_1";b:1;s:7:"level_0";b:1;}}s:11:"contributor";a:2:{s:4:"name";s:11:"Contributor";s:12:"capabilities";a:4:{s:10:"edit_posts";b:1;s:4:"read";b:1;s:7:"level_1";b:1;s:7:"level_0";b:1;}}s:10:"subscriber";a:2:{s:4:"name";s:10:"Subscriber";s:12:"capabilities";a:2:{s:4:"read";b:1;s:7:"level_0";b:1;}}}
Once you have made those changes, you should be able to log in to your WP-Admin and manage your blog as normal.