The nuclear spin systems in CdTe/(Cd,Zn)Te and CdTe/(Cd,Mg)Te quantum wells (QW) are studied using a multistage technique combining optical pumping and Hanle effect-based detection. The samples demonstrate drastically different nuclear spin dynamics in zero and weak magnetic fields. In CdTe/(Cd,Zn)Te, the nuclear spin relaxation time is found to strongly increase with the magnetic field, growing from 3 s in zero field to tens of seconds in a field of 25 G. In CdTe/(Cd,Mg)Te the relaxation is an order of magnitude slower, and it is field-independent up to at least 70 G. The differences are attributed to the nuclear spin relaxation being mediated by different kinds of resident electrons in these QWs. In CdTe/(Cd,Mg)Te, a residual electron gas trapped in the QW largely determines the relaxation dynamics. In CdTe/(Cd,Zn)Te, the fast relaxation in zero field is due to interaction with localized donor-bound electrons. Nuclear spin diffusion barriers form around neutral donors when the external magnetic field exceeds the local nuclear field, which is about $B_L\approx $0.4 G in CdTe. This inhibits nuclear spin diffusion towards the donors, slowing down relaxation. These findings are supported by theoretical modeling. In particular, we show that the formation of the diffusion barrier is made possible by several features specific to CdTe: (i) the large donor binding energy (about 10 meV), (ii) the low abundance of magnetic isotopes (only $\approx$30% of nuclei have nonzero spin), and (iii) the absence of nuclear quadrupole interactions between nuclei. The two latter properties are also favorable to nuclear spin cooling via optical pumping followed by adiabatic demagnetization. Under non-optimized conditions we have reached sub-microkelvin nuclear spin temperatures in both samples, lower than all previous results obtained in GaAs.