Let $K$ be an imaginary quadratic field and $E$ an elliptic curve with CM by the maximal order of $K$, such that $E$ is defined over the Hilbert class field $H$. Is it known whether there is a bound (independent of the degree of $H$) on the order of a $H$rational torsion point on $E$?

Hey, I probably should have answered this one some time ago. It was proved in 1989 by J.L. Parish that the order of an $H$rational torsion point is 1,2,3,4 or 6, and this also can be deduced from work of either Silverberg or PrasadYogananda. In any case the statement you want is at the beginning of section VI in the paper below and the proof is done in section V. http://www.sciencedirect.com/science/article/pii/0022314X89900127 

