I think you have answered the question with your seventh word - organization
You are setting up an organization, not a company, so org.uk is more appropriate.
But as others have said, it seems a free for all with domain names so either will do.
One thing to consider is that "co" domain names are more popular than "org" and if you use the "org" domain someone could grab the "co" domain with the same name as yours which may get confusing.
If you buy the "co" domain people are less likely to try to use the "org" domain.
You could try to grab both and then just have a pointer on the "co" domain that points to the "org" site. I have this on a web site I set up for a friend where they were worried someone would grab the "co" domain.
It is only a few pounds extra per domain.