Interpolating/Decimating CIC Filter Group Delay
Consider an $D$-tap FIR filter with liner phase, the group delay (measured in samples) is $$g=\frac{D-1}{2}\tag{1}$$
and therefore, if it is measured in seconds it will be $$g=T_s\frac{D-1}{2}\tag{2}$$
where $T_s=1/F_s$.
The CIC filter which is also denoted as recursive running sum filter is indeed a special implementation of a moving-average filter. The moving average filter is
$$y[n]=\frac{1}{D}[1+z^{-1}+z^{-2}+\cdots z^{-D+1}]=\frac{1}{D}\sum_{n=0}^{D-1}z^{-n}=\frac{1}{D}\frac{1-z^{-D}}{1-z^{-1}}$$
which can be implemented by a $D$-tap FIR filter. If you compare carefully, the above response is identical to the frequency response of each stage of a CIC filter.
Therefore, we can use $(1)$ and $(2)$ for calculation of the group delay in each stage.
Comparing the response given for moving average with that of the CIC filter, assuming $R$ is rate change and $M$ is the number of samples per stages, then $$D=RM$$ So for $N$ stages we have $$g_{\text{cic}}=N\left(\frac{RM-1}{2}\right)$$ expressed in samples and
$$g_{\text{cic}}=NT_s\left(\frac{RM-1}{2}\right)$$ in seconds.