I have two version of buildroot downloaded. Older version (2014.05) of buildroot needs qtbase-opensource-src-5.2.1. It's OK because the source for this version of Qt has already been downloaded.
But I want to use second version of buildroot (2015.08) for some reason. However, second version needs qtbase-opensource-src-5.5.0. Unfortunately, the computer I use, is not able to download something from internet. So I want to use Qt 5.2.1 with the newer version (2015.08) of buildroot. I cannot figure out how to tell buildroot to use this version of Qt. In menuconfig, I cannot change the version. I cannot find a config or similar file to configure the version of Qt.
Buildroot does not support choosing a specific version of a package, because very often the way to build it will be different for a different version. For Qt5, this is certainly the case.
In your particular case, the easiest is probably to just remove package/qt5 from the buildroot tree, and replace it with a copy of package/qt5 from the 2014.05. Note that this will only work correctly if you use no package that depends on Qt. Also, there may be some changes needed to make it work in Buildroot 2015.08. For example, BR2_PREFER_STATIC_LIB has been renamed to BR2_STATIC_LIBS.
Maybe, however, it's easier to just put qtbase-opensource-src-5.5.0 on a flash drive :-)