1. Likelihood of getting exposed to HIV is zero.
2. You don't have to wait for anytime because you were never at risk.
3. You child is fine, we understand that you are anxious about his or her well being. Let me assure you that there is no risk whatsoever in your situation.
HIV is fragile in nature, it is very difficult for this virus to get transmitted and it happens only in specific situation, Unprotected vaginal or anal intercourse and sharing needles to inject drugs with other users are the only 2 risks.
In your event, the blood present must have been in small amount and was well exposed to the environment, that caused the virus to neutralise, being incapable to infect not only that but also the cut in your body was not deeply gapping that was profusely bleeding for any instance of risk. The fact of the matter is, the whole scheme of believing that the blood was HIV infected person's is an assumption that is derivative of anxiety.
It's understandable, this anxiety must be due to the concern for your little one's well being, once again let me reassure you that he/she is going to be alright since there was no risk present. You don't need to test.
Your situation involves contact with an object that had been in air which is not a risk for hiv. No worries, because you can't get hiv from personal contact except unprotected penetrating vaginal or anal, neither of which you did. Analysis of large numbers of infected people has proven that people don't get hiv in the way you are worried is a risk.
HIV is effectively instantly dead in air so the worst that could happen is dead virus rubbed you, and that which is dead cannot live again so you are good. Blood and cuts would not be relevant in your situation since the hiv has become effectively dead, so you don't have to worry about them to be sure that you are safe.