I'd love to be able to give a single figure answer to the this question, but the truth is; costs vary from project to project depending on client needs and style considerations.
As a very rough ballpark figure, projects tend to start from around £3000. This sort of budget allows for some very simple iconography and illustration, a short script equating to around ~30 seconds of animation, a voiceover provided by myself, and a stock music track.
If you're just after something small like a logo animation, the cost of this depends on the complexity of the logo and how easy it is to animate. Some logos lend themselves well to animation, others need a little more thought! Smaller projects like this range from around £750 - £1500+, again this is just a very rough ballpark figure and would vary from project to project.
Animation, like any other field of design takes time to get right. Usually the more time and budget available: the greater the scope of possibility. With a higher budget, you can look to include more features such as more detailed illustration work, 3D elements and animation, sound effects, even a fully blown script writer to produce your script for you.
That's not to say that every animation project needs a decent budget to be good, there's plenty that can be done towards the lower end of the budget spectrum, it just might require a different approach and some very creative thinking.
Hopefully that covers the basics on how much an animation costs and what you can expect, for more advice on budgets and costings, get in touch via [email protected] and we can have a chat about what's best for your project!