Not discouraging you from seeing a medical professional as that is definitely the best course of action (doesn't need to be a doctor, a practice nurse should be able to advise - I'm not sure what the equivalent is the the US, sorry!) but I have had deeper cuts on my thumbs that healed fine. I think I injured nerves both times though as I had a lot of neuropathic pain afterwards (basically, a very sharp stabbing/burning pain whenever I brushed against my thumbs even lightly that lasted for months), but I'm not sure if seeing a health professional would have prevented that.
If you choose not to see anyone about it now, if it get more painful or red or swollen, or if there is any pus coming out of the wound, you should definitely speak to someone as those might be signs that the cut has gotten infected and that you might need some antibiotics.
Hope it heals soon!