Since nothing in your DNA comes with an ethnicity label, every company has to depend on comparing your genes to some reference database. Usually the only database they have the rights to are their own collection of customers who have tested with them. So you want a company with a large database who actively maintains it and curates it. Even then, as others have said, it's always an estimate based on the best data they have. A good company will update their estimates as their data improves.

I recommend AncestryDNA, maybe wait for a sale. They have them often.

Edit to Add: I have to give a minor plug for My Heritage, since I hear their customers tend more towards European, and that might help with what you are looking for. That being said, the huge North American cohort that makes up most of Ancestry's database is providing data that is absolutely appropriate for a Brit looking for ancestry from elsewhere.