Finding the perfect web design agency can take a little bit of time. There are hundreds of web designers based throughout the UK, and a quick Google search will demonstrate this if you type in “web design + your town”! You will probably need to take some time to go through the websites of the different agencies you find in your local area until you find one you really like the look of. You should be able to see from their own website the sort of work they produce. If not, try looking for a portfolio or past projects page. Most companies will show off past websites they have worked on, so you can get an idea of how they work.
One thing to bear in mind when looking for a web designer is that not all web design companies offer the same skill set – some designers work using different types of code, different CMS and different languages for programming. It may be that what you require needs a specialist skill, so be sure to check with them what skills they use.