To assume office, a senator or representative needs a certificate of election signed by the governor.
After voting is completed on January 5, 2021, the county election boards meet to canvass and certify the votes for each county. Those certifications are sent to the state board. The state board certifies the results for the state. Once that is complete, the governor issues the certificates.
If a race is close or contested, there could be a delay of days or weeks for recounts or other challenges.
If elected, when would Ossoff and Warnock take office?
There is no fixed date for the governor to issue the certificate of election, a week after the election would not be unreasonable; though 10 to 14 days may be more typical and, absent problems, not more than 17 days.
The National Association of Secretaries of State (NASS) provides a reference for states' laws for canvassing election results. For Georgia, the schedule is —
County officials canvass local election returns and transmit results to the Secretary of State no later than the second Friday after the election. The Secretary of State must canvass and certify election results for state and federal offices no later than 17 days after the election. [...] (Georgia Statute 21-2-493; 499)
Specifically, Georgia Code Title 21. Elections, Chapter 2, Article 12 Returns, uses the imperative "shall immediately" for both the county and state boards to complete the canvass and certify the results; meaning there is no requirement to wait until the second Friday to forward the county certification, nor wait 17 days for state certification.
Then, in Georgia Code Title 21. Elections § 21-2-502,
(b) United States senators; representatives in Congress; members of the General Assembly.
(1) Upon completing the tabulation of any election for United States senator or representative in Congress, the Secretary of State shall lay the same before the Governor, who shall immediately issue certificates of election and commissions under the seal of the state, duly signed by the Governor and attested by the Secretary of State and deliver the same to the candidates receiving the required number of votes to be elected to the respective offices.