How is the nilpotent of class 2 (nil-2) free product of groups defined?

I came across this construction reading the following paper.

Alan H. Mekler (1981). Stability of nilpotent groups of class 2 and prime exponent . The Journal of Symbolic Logic, 46, pp 781-788. doi:10.2307/2273227.

There the author use the nil-2 free product in order to build the nil-2 exponent $p$ group generated by some nice graph $\Gamma$.