If you have a traditional balance in your TSP account, the taxes on those contributions (and their earnings) are deferred until the money is paid to you. Therefore, the TSP annuity payments comprised of traditional amounts will be taxed as ordinary income in the years when you receive them.
If you have a Roth balance in your TSP account, those contributions were made after-tax. The TSP annuity payments comprised of Roth contributions will not be taxed. Whether the Roth earnings portion of any annuity payment is taxed depends on whether that particular payment meets the IRS rules for qualified earnings.
Uniformed services TSP accounts may include contributions made from pay earned while serving in a combat zone. Certain pay earned in a combat zone is exempt from Federal income tax.
If you have tax-exempt money in the traditional balance of your uniformed services TSP account, and you choose to purchase an annuity, the annuity provider will calculate the amount of tax-exempt contributions that will be paid as part of the traditional portion of your annuity payment and will inform you of this amount. The tax-exempt portion of your payments will be spread out based on your life expectancy (and that of your joint annuitant, if applicable).
Your annuity payments are not subject to the IRS early withdrawal penalty tax regardless of how old you are when they begin.