Firstly, the RIBA holds a register of every chartered practice in the UK, and the user friendly find an architect website helps you find a chartered architect in your area. The site lists practices within a set number of miles of your location, and each practice has a profile page with a short description.
Websites such as Houzz.co.uk are also great starting point for residential architects. Here you can browse photographs of finished projects to get a feel for each practice's style, and read a summary about the practice on their profile page. Other websites such as Design-for-me offer a good range of design-led architects and designers, and allow you to invite 3 designers who have shown an interest in your project for a free consultation.
There are a number of other professional services websites available that allow you to post a short description about your job and get contacted directly by an architect or designer. However these websites also feature architectural technicians and CAD drafts-people and the level of service on offer will vary greatly. For example many online services don’t come to your house for a site visit, and many won’t see the project through on site. Always be clear what it is you want and the check the level of service is being offered. Also beware, non-chartered architects do not adhere to any of the professional regulations that registered architects are required to and there is no guarantee of good design!
And don't forget, pre-digital methods still exist! Adverts in local or home and lifestyle magazies, such as the South East London Journal are more likely to feature adverts which are specific to your area.