For the benefit of the OP.
A brick chimney is easily recognised by a chimney stack, with either a pot or gas terminal on your roof, which relies on the natural circulation of air through the room and up your chimney to expel the products of combustion. The minimum internal diameter will be 7” (or greater) and it may have had a real solid fuel fire in previously, especially in the case of older properties sometimes identifiable by an existing chair brick at the back of the fire opening.
Note that your chimney may or may not have a terminal(pot) this would be part of advice you need on how to remedy.
Also, your chimney looks to be of stone blocks and brick. It’s still a class one. Totalshell was correct on that score.
If you read up and are a quick learner, there’s no reason you couldn’t show yourself to be competent to building control and complete a safe installation. If you have any doubts, call someone in. There’s more to know than is worth trying to list here.
Its not cheap, but it’s worth it being done right.
Incidentally, £40 to test the integrity of flue seem a little low to me. I wouldn’t do it for that. It’s not worth the trip out.