The £5 increase will come into force in April, and is significantly less than the £10.50 rise that was applied earlier this year.
The government has returned to calculating the increase by using an annual inflation rate rather than a monthly one, which was used last year.
The cost of a TV licence is due to increase in line with inflation each year until 2027, following an agreement made in 2022.
Foreign Secretary David Lammy has also announced an extra £32.6m for the BBC World Service in 2025-26.
Meanwhile, the review of the BBC’s royal charter will include a public consultation enabling audiences and stakeholders to give their views, the government said.
Nandy said: “The BBC provides much-needed programming for households across the country, including children’s education, world-class entertainment and trusted news for all people in all parts of the UK. I want to see it thrive for decades to come.
“Through the Charter Review, we will have an honest national conversation about the broadcaster’s long-term future, ensuring the BBC has a sustainable public funding model that supports its vital work but is also fair and responsive to those who pay for it.
“In the short term, we are providing the BBC with funding certainty, while supporting thousands more households facing financial hardship to spread the cost of a TV licence.”