In Italy most of blood tests are done under the national health system, both in public and private labs. You most likely will be asked a prescription, even if you do not qualify for the NHS.
In Florence, a fast way could be addressing the medical practice of Misericordia di Firenze (an historic first aid association) at Vicolo degli Adimari 1 (few meters from Piazza del Duomo); they take blood samples and send them to a private lab that will deliver results either on paper or online. If you know exactly what you need no prescription is needed. Try writing them at [email protected] and ask if they do INR tests.
(By the way, this is an excellent address if you need an extra specialist visit.)