Welcome to the forum. I'll try to help.
Initial response, after reading only the first sentence: STDs are not transmitted by fingering. If that is your only exposure to a sex partner, there is no STD you need to worry about.
Now I have read the rest. Having a wart on your finger does not increase the risk of STD. In terms of your partner's health, the HPV types that cause common warts of the hands and feet (most commonly, HPV-1) do not infect the genitals. They are entirely different strains than those that cause genital warts (HPV-6 and 11) and cervical cancer (HPV 16, 18, 31, 34, 45, and several others).
Bottom line: No worries here for either you or your partner.
Best wishes-- HHH, MD
Thank you.