I think the chance of anything passing is very low for a number of reasons. The person would need to have something like HIV or hepatitis B & C for there to be any risk in the first place. Also the person would need to have this blood on a part of the nail that would enable it to still be there after it had broken your skin. Given this I think the risk is very low. But I can see if someone did have one of these infections and the conditions were exact maybe it could be possible.
This is the good thing with going for a test as this will allow you to remove all doubt if this is still concerning you. You could go for a HIV and hepatitis B & C test, we find most infections would show up on a test 6 weeks after they had passed to someone, but still recommend a final test at 3 months to make sure.
Let us know if this does not answer your question or if you have any more questions or concerns.