The health and happiness bonuses (as well as any other bonuses the resource gives) apply to all cities connected to the trade network the resource is connected to. In some cases I find it better to get a resource from outside my cities workable areas, as I don't think eg. Winery is that good an improvement - I'll rather build a cottage, watermill, or something else on the tile if possible.
Note that the resources have two effects: added food / production / commerce on the tile and a globla effect of added health or happiness or strategic effect (most often ability to build some types of units, but may also speed wonder building, provide fuel for powerplants, and so on). The global effect is what you get in all cities, the F/P/C bonus only applies to the tile should it be worked.
Hehe, I got there first for once

(but only because Elandal took the time to add a load of other useful info).