You do not need to worry about getting HIV this way. HIV is transmitted by unprotected sex or direct blood-to-blood contact. The HIV virus dies within seconds after leaving a person’s body. So, HIV from someone else’s blood needs to get directly into your veins to cause an infection. It sounds to me like the two small cuts on your hands were healed up (no longer bleeding) and so HIV could not get into your bloodstream this way.
It is possible to get a Hepatitis C infection this way, as the Hepatitis C virus can live longer outside the body. Again, this is not likely if the cuts were healed up.
If you are still concerned, you could get an HIV and a Hepatitis C test 3 months after this happened to be reassured.
I hope that helps answer your question,