As with your common experience, heat is transferred from areas of high concentration (i.e. a hot object) to areas with less concentration (i.e. anything that is less hot). So these carbon nanotubes would transmit the heat to the air around it, to the material behind them, and also the heat would conduct along the nanotubes.

Copper is very conductive, so it is a good idea to transfer heat using copper. However, heat is not going to flow from an area of low concentration (i.e. low temperature) to an area of high concentration (i.e. high temperature). In the same way that water does not flow up hill, heat will not flow from cold to hot. Because the hot tub is always hotter than the swimming pool, any sort of system that connects the two will always heat the swimming pool, and cool the hot tub.