Suppose that N is the CMC hypersurface which is \epsilon $\epsilon$ close to S. Take the tube of radius \epsilon $\epsilon$ around S. One can check that this has (typically variable) mean curvature on the order of 1/\epsilon. $1/\epsilon$. If one chooses this tube correctly, it is tangent to N and this gives a lower bound for the mean curvature of N at the point of contact. Since the mean curvature of N is constant, this means that it is everywhere bigger than 1/\epsilon.$1/\epsilon$.
The situation is now a bit better understood than when that paper was written. First, Rosenberg proved a theorem that if N is a surface in a 3-manifold and has mean curvature H and H is very large, then one of the two regions N bounds in the ambient manifold has inradius less than C/h.
Second, it is only true that the set on which the N_t $N_t$ collapse must be minimal if the norm of the second fundamental form of N is comparable to the mean curvature. If this fails -- and a typical scenario is a sequence of spheres joined by very small necks, then it seems to be the case that the collapsing set is not necessarily minimal.
Finally, regarding your question -- by the same comparison result (using geodesic tubes) it is certainly the case that H is increasing as N_t $N_t$ collapses, so this gives some sort of average increase of H'(t). There maybe could be some exotic examples where the mean curvature wobbles and H' is not necessarily tending to +\infty. $+\infty$.
There are lots of other good questions here -- Harold Rosenberg had several questions which remain unanswered. For example, if T is a `geodesic triangle' in a 3-manifold, then is there a CMC tube which condenses to T? The problem is whether it is possible for the CMC surfaces to turn the corners at the vertices of T.