My two penn'orth, fwiw:
It's unreasonable to expect the dealer who did the health check to pay to replace the worn tyres, however...
Most (all?) of us here are car enthusiasts and have some understanding of how our cars work. Many drivers are not, don't have a clue how to do any more than add fuel, and rely 100% on what a garage tells them. If a garage is going to carry out a "health check" that reports on the safety/legality of a vehicle then it should be carried out with proper care and diligence. I completely understand Wolfie1's point regarding allowed times for jobs and bonus schemes (my father used to be a motor fitter and later in his career a garage workshop manager), but that doesn't absolve the garage from culpability if they give incorrect advice.
In the circumstances, while it's impossible to know if cords were visible on the OP's tyres when they were allegedly inspected as part of the "health check" 900 miles previously, it's not unreasonable to assume that the tyres exhibited significant wear on the inner shoulders that could have been seen and should have been reported. I would be inclined to have a polite conversation with the Service Manager to express my disappointment that it was not, and that you had relied upon the "health check" report to assume that the tyres were safe when they were at best borderline and at worst illegal. We can assume that the SM has a brain, so they will understand the significance of what you are telling them. Whether they choose to act upon it is, of course, another matter.