To do this we set up permanent redirects from the website root to your new site.
Get your list of backlinks and order the list with the ones you highlighted as important at the top, to those you have not considered relevant afterwards. Remove all columns except for the URLs, rename the column "old" and add a new column next to that with the header "new".
For each URL you should enter the URL of a page on your new site.
Now remove the domain names and http:// cruft. Example:
Http://www.gwp-caribbean.org/iwrm/news.html http://www.gwp.org/en/caribbean/news-and-activites
becomes
/iwrm/news.html /en/caribbean/news-and-activites
You may find that there is no directly equal page, so many may want to go to the home page or a page that is the same subject but not identical. Ex. an old news item (if very old) may want to point to your main "News" page instead of another article.
Next you should provide this list to the GWP webmaster, who will check it for any conflicts with existing redirects and include it into the redirects.csv file at the site root.
If there are any conflicts found you will need to contact the sites linking to these pages and ask them to change their link to the new page on the new site themselves.