New bootloader on Nano

Tools -> Board -> Board Manager. Locate the official AVR board package on that screen and update it.

You do not need to enter any URL - the official AVR board package will always show up there.

If you don't see an old bootloader option, you don't have 1.6.21 of the AVR board package.

Tools -> programmer is ignored unless you are burning bootloader or doing upload using programmer, neither of which is what you want to do. When doing this, you will know what option to select from this menu, because it is a physical piece of hardware needed for those operations. It is ignored when doing normal uploads (on official arduino boards - on third party board packages that support parts without a bootloader, it is used for all uploads).