Unfortunately there are people in this world who are happy to take up other peoples professional time and try to get away without paying. It quite simply boils down to the fact that if you're working as a freelancer you absolutely have to at the bare minimum take a deposit of at least 25% and really, you need to be using some of project agreement or contract, stating exactly what work you're doing and how much they're paying you for it. It's when you don't do these things that people think they can get away with not paying you, because you've given them the deliverables and there is no proof of any agreement between you and them, you've effectively given up any position of power in that situation and they know this.
Maybe they have genuinely forgot, or are rushed off their feet with something and you'll get the payment in a week from now. But imagine if this was a large project that you were relying on to pay your bills... I would honestly start getting into the habit of taking a deposit and working with a basic project agreement for every project you work on.
As for what you can do now, well it seems to me like you have three options. Keep contacting them and hope for the best, let it go and put it down as a learning experience or chase them up for it in the small claims court. I would imagine that you don't have the inclination, time or the finances to pursue them legally, and really it's not going to be worth it. You could just let it go, and put it in your portfolio as charity work, which always goes down well but if it was me, I would be wanting payment if that's what they promised.
Unfortunately there is every chance that you won't get anything, if they are really determined not to pay you but even if it's just for the principle it's worth spending a bit of time on it, just in case. You can always try visiting them locally, because they can't ignore you then and a lot of people can't deal with confrontation when they know they're in the wrong, but this approach does rely on you knowing what to say and how to say it.
If you create any art or design work, you are automatically granted the status of copyright owner, you do not need to sign up to or register anything. This means that they are infringing on your copyright by using it without your permission. Another course of action would be to send a cease and desist letter. A lot of the time people think they don't need to pay you, because they believe you're small, lacking in knowledge and don't know how to react in these situations, so they think they can get away with it. A professionally worded c&d letter, if this is the case, will certainly make them think twice about not paying you. Although you need to be careful because a badly worded c&d letter could do you more harm than good.
Personally I would not expect a charitable organisation to simply decide not to pay you, and so if they continue to ignore your emails and calls then go and pay them a visit, personally, with the intention of finding out why they haven't paid you. Some times people just disappear rather than owning up to something, such as they don't have the money to pay you just yet, or something else could be going on. I assume all your contact has been via email so far?