I bank with The Co-operative Bank (who are the only UK bank with an ethical investment policy who do current accounts), and they do a Community Banking account specifically for non-profits http://www.co-operativebank.co.uk/servlet/Satellite/1196151412581,CFSweb/Page/Business-CommunityBanking (which also allows you to apply for grants from the bank towards worthy causes). When I applied for one of their Community Banking accounts, sorting out the anti-money-laundering ID requirements for all the signatories did take some time, but those are statutory requirements you will have to fulfill when applying for any UK account.
Yours, Joe. (User:Joeblakesley)