According to information of Georgia Visa Portal, you don't need a visa as long as you visit Georgia during validity period of your UK visa and for no longer than 90 days in consecutive 180 days.
Holders of valid visa or residence permit of the country you have selected are exempted from visa requirements. You can enter Georgia without a visa for 90 days in any 180-day period. Please, note that you must present relevant valid visa or residence permit along with your travel document/passport at the moment of crossing Georgian border.
Detailed information on visa free travel is available on the following website: www.geoconsul.gov.ge.