There are 1,174 flooring companies listed on this site, and reading through them is a tour of how the trade actually organises itself, which is not how customers usually imagine it.
The listings sort into three broad kinds of business. Retail-led firms, showrooms and shops, sell the floor and arrange fitting through employed or partnered fitters. Fitting-led firms are the reverse: fitters first, who can supply through trade accounts but are happy to lay a floor you bought elsewhere. And restoration specialists work on the floor already in the house, sanding, refinishing and repairing rather than replacing. Plenty of companies span two of these, but a listing usually makes the centre of gravity clear within a line or two.
The practical use of this: match the kind of firm to the kind of job. Buying boards and a fit together gives you one throat to choke if something fails, valuable with warranties. Supplying your own floor and hiring a fitting-led firm can work out cheaper, but read your board warranty first, because some require professional fitting to specific standards. And if the house has original boards, talk to a restorer before either of the others, replacement is often the answer to a question nobody asked.
Showrooms anchor to premises, so they cluster in towns and serve the radius a customer will drive. Fitters travel further and describe coverage in postcode spans. Restorers, whose kit fits in one van, often range widest of all. This is worth knowing when the nearest showroom search comes up thin: the fitter or restorer who covers you may be based two towns over and entirely used to the drive.
Start from your local listings and read for the centre of gravity: supply, fit, or restore. Then describe the job once, naming the floor you have and the floor you want, and let the firms whose trade matches reply. The mismatched ones will filter themselves out, which is the directory doing its job.