You are probably fine since you removed it. I don't think you need to see a doctor unless you have some symptoms develop. Something like a yeast infection or vaginal bacterial infections of BV are not uncommon when this happens. Toxic shock is serious but again, very doubtful at this point that this is a concern as tampon is now removed and you are seemingly fine. good luck
I would see a doctor just to be sure. On the side of the box or on the back it says to keep tampon in for a maximum of 8 hours because of TSS (Toxic Shock Syndrome) which could make you very sick and could also be deadly. It sounds to me that your fine and you haven't had any discomfort or pain but I would still get yourself checked out just to be on the safe side. Hope this helps.