The blood thinners you speak of would reduce your platelet count so the initial injury, if bleeding, would want to keep bleeding due to the thinned blood. However, it is your white blood cells that do the healing and they are not affected by blood thinners. So I wouldn't be concerned about it unless you observe that it is taking a REALLY long time to heal.
Yes it affects the healing process it takes about 5-7 days for the effect of blood thiners to wear out of your body its also very serious when you take blood thiners because of the risk of stomach bleeding or in your case cuts that will no heal bc the blood thiners keep the blood from cuagulating thats why people with blood cloths take this pill.