Updated
The two routes to sign-off
If the installer is HETAS registered for that type of work, they can self-certify it. They notify the installation through HETAS, and you receive a HETAS Certificate of Compliance in the post. No separate visit from the council is needed.
If they are not registered, the work has to go to your local authority instead, as a Building Notice. Building Control then inspect it and issue the completion certificate, and there is a fee for that.
Both routes are legitimate. The difference is who signs the work off, and it is worth asking any fitter which one you are getting before they start. We are HETAS registered, so our installations are self-certified.
Why the certificate matters
The record of the installation is what your home insurer and, later, a buyer's solicitor will ask for. A stove with no certificate behind it can hold up a house sale, and can leave you arguing with an insurer after a fire.
What the regulations actually cover
The main document is Approved Document J, which covers combustion appliances: the air supply the stove needs, how the smoke is carried away by the flue, the hearth and the distance to anything that can burn.
It also requires a carbon monoxide alarm in the room where a fixed solid fuel appliance is installed. That goes in as part of the job.
And when a hearth, fireplace, flue or chimney is provided or extended, a durable notice plate has to be fixed in the house recording what it is built for, so whoever works on it next knows what they are dealing with.
What to ask before you book
Are you HETAS registered, and can I check you on the HETAS register? Will I get a Certificate of Compliance, or do I need to apply to Building Control myself? Is the carbon monoxide alarm included? Does the price include sweeping and checking the flue first?