Congress rejected President Andrew Johnson's readmission of the Confederate States to the Union. Congress also refused to seat the House and Senate members elected by those Confederate States. When and how were those Confederate States and their elected officials eventually accepted by Congress?

Post-Civil War Reconstruction was placed under control of Congress in 1866. The relevant actions that Congress took included denying congressional seats to the former Confederate states, in part in reaction to the passage of “Black Codes,” which imposed legal restrictions on African Americans, and the widely evident violent attacks on black citizens. Moreover, Congress passed the Civil Rights Act of 1866 and initiated the 14th Amendment, which then underwent the ratification process. Then, in 1870, Congress began the process to add the 15th Amendment to the Constitution. The 14th Amendment guarantees citizenship and equal protection to African Americans, as well as reducing congressional representation to the states that refused African American voting. The 15th Amendment guaranteed voting rights to African American men (but not women).
Ratification of the 14th Amendment was one requirement for re-entry, which many states initially refused to do. In 1867, Congress attached the re-entry into Congress to the states’ adoption of constitutions that matched the guarantees of the 14th and 15th amendments. This would be enforced, in part, thorough the Military Reconstruction Act, which placed those states under martial law in military districts. Thus, the process of re-entry advanced state by state, depending on when it created a new constitution. The lengthy process also included the impeachment of President Johnson, who retained office by a single vote. In 1868, Ulysses S. Grant, who had resigned as Johnson’s Secretary of War, was elected president.
The final states were thus readmitted under the Grant administration. As they achieved the necessary compliance, they were readmitted over a period of four years, from July of 1866 through July of 1870. On July 24, 1866, Tennessee was the first former Confederate state readmitted to the Union. In June of 1868, Arkansas, Louisiana, Florida, North Carolina, and South Carolina were all readmitted, followed by Alabama in July. In 1870, when the 15th Amendment was also ratified, the last four states were readmitted: Virginia (January), Mississippi (February), Texas (March), and on July 15, 1870, Georgia became the last state readmitted.
http://www.digitalhistory.uh.edu/disp_textbook.cfm?smtID=2&psid=3103

https://www.pbs.org/wgbh/americanexperience/features/reconstruction-timeline/