How long after the Big Bang would CMB radiation have been between 273 and 373 K?
The temperature of the cosmic microwave background scales as the inverse of the cosmic scale factor $a$. i.e. When everything was at half the separation it is now, then the CMB was twice the (absolute) temperature. The scale factor in turn is reciprocally related to the redshift by $a/a_0 = (1 +z)^{-1}$, where $z$ is the redshift and $a_0$ is the present-day scale factor, usually taken to be 1.

From this we see that
$$T(z) = T_0 (1+z),$$
where $T(z)$ is the temperature at redshift $z$ at some time in the past and  $T_0 = 2.73$K is the temperature of the CMB now. For the range of temperatures you specify requires redshifts in the range $99&lt; z&lt; 136$. 

Getting from a redshift to a time in the universe since the big bang is not so simple. The answer depends on what you take to be the "cosmological parameters" - i.e. the values of the cosmic matter density, the dark energy density and so on.

However, we can avoid looking "under the hood" and use a cosmology calculator. The one I have linked to is for a "flat" universe and has default values for what are currently good estimates for the matter density and present-day Hubble parameter. For the range of $z$ I found above, this corresponds to an age of between 10.8 and 17.3 million years after the big bang.

I guess you are perhaps thinking about these ideas that life could have been around about 15 million years after the big bang.